Bilinear Maps and Tensor Products in Operator Theory [1 ed.] 9783031340956, 9783031340932

Table of contents :
Preface
Contents
1 Linear-Space Results
1.1 Notation, Terminology and Definitions
1.2 Extension of Linear Transformations
1.3 Quotient Space
1.4 Additional Propositions
2 Bilinear Maps: Algebraic Aspects
2.1 The Linear Space of Bilinear Maps
2.2 Extension of Bilinear Maps
2.3 Identification with Linear Transformations
2.4 Additional Propositions
3 Algebraic Tensor Product
3.1 Tensor Product of Linear Spaces
3.2 Further Properties of Tensor Product Spaces
3.3 Tensor Product of Linear Transformations
3.4 Additional Propositions
4 Interpretations
4.1 Interpretation via Quotient Space
4.2 Interpretation via Linear Maps of Bilinear Maps
4.3 Variants of the Linear-Bilinear Approach
4.4 Additional Propositions
5 Normed-Space Results
5.1 Notation, Terminology and Definitions
5.2 Extension of Bounded Linear Transformations
5.3 Normed Quotient Space
5.4 Additional Propositions
6 Bounded Bilinear Maps
6.1 Boundedness and Continuity
6.2 Identification with Bounded Linear Transformations
6.3 Extension of Bounded Bilinear Maps
6.4 Additional Propositions
7 Norms on Tensor Products
7.1 Reasonable Crossnorms
7.2 Projective Tensor Product
7.3 Injective Tensor Product
7.4 Additional Propositions
8 Operator Norms
8.1 Uniform Crossnorms
8.2 Tensor Norms
8.3 Dual Norms
8.4 Additional Propositions
9 Tensor Product Operators
9.1 Tensor Product of Bounded Linear Transformations
9.2 A Hilbert-Space Setting
9.3 Some Spectral Properties
9.4 Additional Propositions
References
List of Symbols
Author Index
Index
Index

Citation preview

Universitext

Carlos S. Kubrusly

Bilinear Maps and Tensor Products in Operator Theory

Universitext Series Editors Nathanaël Berestycki, Universität Wien, Vienna, Austria Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain John Greenlees, University of Warwick, Coventry, UK Angus MacIntyre, Queen Mary University of London, London, UK Claude Sabbah, École Polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France Endre Süli, University of Oxford, Oxford, UK

Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal, or even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may find their way into Universitext.

Carlos S. Kubrusly

Bilinear Maps and Tensor Products in Operator Theory

123

Carlos S. Kubrusly Professor Emeritus Catholic University of Rio de Janeiro Rio de Janeiro, Brazil

ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-3-031-34095-6 ISBN 978-3-031-34093-2 (eBook) https://doi.org/10.1007/978-3-031-34093-2 Mathematics Subject Classification: 47A80, 47A07, 46M05, 46A22, 54C20, 15A63 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Jessica and Alan

Istigkeit—wasn’t that the word Meister Eckhart liked to use? ‘Is-ness.’ The Being of Platonic philosophy—except that Plato seems to have made the enormous, the grotesque mistake of separating Being from becoming, and identifying it with the mathematical abstraction of the Idea. He could never, poor fellow, have seen a bunch of flowers shining with their own inner light and all but quivering under the pressure of the significance with which they were charged; could never have perceived that what rose and iris and carnation so intensely signified was nothing more, and nothing less, than what they were—a transience that was yet eternal life, a perpetual perishing that was at the same time pure Being, a bundle of minute, unique particulars in which, by some unspeakable and yet self-evident paradox, was to be seen the divine source of all existence. Aldous Huxley’s The Doors of Perception

Preface

This book, intended as a first course in bilinear maps and tensor products, introduces tensor products in Banach spaces to graduate students, helping them to learn the subject from the beginning. Tensor products, particularly in infinite-dimensional normed spaces, are heavily based on bilinear maps. The book brings these issues together by using bilinear maps as an auxiliary, yet fundamental, tool for accomplishing a consistent, useful, and straightforward theory of tensor products. Bearing in mind that graduate students typically find this subject complex and challenging, I have presented the material in ways that should make grasping the concepts simpler and easier. Nine chapters, each consisting of four sections, make up the book. Although the text is not formally split into parts, it has been naturally divided into two implicit portions. The first offers a purely algebraic introduction to bilinear maps and tensor products of linear spaces, while the second brings these concepts to the realm of normed spaces and therefore to Banach spaces, with a few ventures into Hilbert spaces. In order to keep the prerequisites as modest as possible, there are two introductory chapters, one on linear spaces (Chapter 1) and another on normed spaces (Chapter 5), summarizing to a large extent the background material required for a thorough understanding and efficient learning strategies—and also establishing a fixed scheme of notation and terminology. However, an introductory course in functional analysis will naturally be assumed. Chapter 1 supplies those essential algebraic notions necessary in subsequent chapters, and only those. These are all related to linear spaces only, mainly focusing on extension of linear transformations and on quotient spaces of a linear space modulo a linear manifold. Algebraic aspects of bilinear maps are presented in Chapter 2. These first two chapters lay the groundwork required to introduce an axiomatic theory of algebraic tensor products of linear spaces, which is done in the subsequent chapters. Chapter 3 introduces a unified approach to algebraic tensor products, where the universal property is taken as an axiomatic starting point, rather than as a theorem for a specific

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Preface

realization. This leads to an abstract notion of algebraic tensor products of linear spaces. Concrete realizations of tensor product spaces, as interpretations of the axiomatic formulation, are considered in Chapter 4. Two approaches to offer such interpretations are investigated, namely, the quotient space approach and the linear maps of bilinear maps approach, and in these approaches, examples and embeddings of tensor product spaces are exhibited. These first four chapters on algebraic concepts comprise the first part of the book. Chapter 5 contains the essential normed-space theorems required in the book. It may be thought of as the normed-space counterpart to the linearspace outline of Chapter 1. Chapter 5 deals mainly with extension of bounded linear transformations and with normed quotient spaces, whose purpose is to put together most of the results (and only those results) that will be necessary farther along in the book. Normed-space aspects of bilinear maps are treated in Chapter 6, where continuous bilinear maps are considered, emphasizing extension theorems for bounded bilinear maps. Chapters 5 and 6 together allow us to advance an axiomatic theory of tensor products of Banach spaces. Chapter 7 contains a detailed account of basic results on tensor products of Banach spaces, featuring suitable norms that equip a tensor product space of normed spaces. By suitable norms, we mean what are commonly referred to as reasonable crossnorms. The chapter concentrates on the notions of two extreme instances of reasonable crossnorms, viz., the injective and the projective norms, leading to the definitions of injective and projective tensor products. Chapter 8 focuses on the central idea of a uniform crossnorm, which is essentially a reasonable crossnorm that makes the tensor product of bounded linear operators a bounded linear operator itself, satisfying an operator-normlike property. A feasible theory of tensor products of operators is based on the concept of uniform crossnorm. This is dealt with in Chapter 9, and includes the Hilbert-space tensor product of Hilbert spaces, and the associated Hilbertspace tensor product operator made up of Hilbert-space operators. Some aspects of their spectral properties are also discussed in this final chapter. Each chapter has a common closing section entitled Additional Propositions. It consists of (i) auxiliary results required in the sequel, and (ii) additional results extending the subject beyond the bounds of the main text. These are followed by a set of Notes, where each proposition is briefly discussed and references are provided indicating proofs for all of them. Such Additional Propositions also play the role of a section of proposed problems, and their respective Notes can be viewed as hints for solving them. This section is followed by a list of Suggested Readings that offer alternate approaches, different proofs, and further questions related to the subject of the chapter. The contents of the book have been devised to be covered in a one-semester graduate course (but as usual the pace of lectures will vary according to specific needs). Since the prerequisites are kept to a minimum, the book is rather self-contained, and so, while it is directly addressed to mathematicians, it

Preface

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may be accessible to a wider audience of students in mathematics, statistics, economics, engineering, and physics. It may also be a useful resource for mathematicians working in areas that deal with tensor products, as well as for scientists wishing to apply tensor products to their fields. In order to attend to a continual request from students, many proofs here are given in great detail, and I do hope this will not be overly boring for the experienced mathematician. I believe complete and detailed proofs, some even emerging from first principles, will help students who are having their first contact with the subject; and this also includes occasional discussions on delicate points, trying to explain some usually hidden features. The logical dependence of the various sections and chapters is linear. An updated bibliography is given at the end of the book, listing only references cited within the text, which contains most of the classics and also recent texts reflecting some of the present-day techniques. That being said, let me also say that the book was not designed to provide a comprehensive treatment on all aspects of tensor products of Banach spaces; much less is it intended to cover the celebrated Grothendieck’s theory. On the contrary, as already mentioned, it is a book for a first course, and it intends to open the doors for further readings. For instance, it may open the way towards a reading of the excellent texts by Ryan [83], Diestel, Fourie and Swart [17], and Defant and Floret [15], perhaps in this order, whose works have been crucial for compiling the present text. Along these lines, it is equally fair to register here that I benefited from the help of highly qualified referees who suggested significant improvements for the final version of the book, and I am really grateful to them. Thanks are due to the copyeditor Brian Treadway as well, who did an admirable job. I also wish to thank Elizabeth Loew from Springer New York for a pleasant and lasting partnership. All in all, the book tries to take the students by the hand to the doors of perception, so they can push the doors open. I wish them a nice trip. Rio de Janeiro April 2023

Carlos S. Kubrusly

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Linear-Space Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Bilinear Maps: Algebraic Aspects . . . . . . . . . . . . . . . . . . . . . .

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2.1 2.2 2.3 2.4

Notation, Terminology and Definitions . Extension of Linear Transformations . . Quotient Space . . . . . . . . . . . . . . . . . . Additional Propositions . . . . . . . . . . . .

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The Linear Space of Bilinear Maps . . . . . . Extension of Bilinear Maps . . . . . . . . . . . . Identification with Linear Transformations . Additional Propositions . . . . . . . . . . . . . . .

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3 Algebraic Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Tensor Product of Linear Spaces . . . . . . . . . . Further Properties of Tensor Product Spaces . Tensor Product of Linear Transformations . . . Additional Propositions . . . . . . . . . . . . . . . . .

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4 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Interpretation via Quotient Space . . . . . . Interpretation via Linear Maps of Bilinear Variants of the Linear-Bilinear Approach . Additional Propositions . . . . . . . . . . . . . .

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Contents

5 Normed-Space Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4

Notation, Terminology and Definitions . . . . . . . Extension of Bounded Linear Transformations . Normed Quotient Space . . . . . . . . . . . . . . . . . . Additional Propositions . . . . . . . . . . . . . . . . . .

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6 Bounded Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1 6.2 6.3 6.4

Boundedness and Continuity . . . . . . . . . . . . . . . . . . Identification with Bounded Linear Transformations Extension of Bounded Bilinear Maps . . . . . . . . . . . . Additional Propositions . . . . . . . . . . . . . . . . . . . . . .

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113 117 119 128

7 Norms on Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.1 7.2 7.3 7.4

Reasonable Crossnorms . . . Projective Tensor Product . Injective Tensor Product . . Additional Propositions . . .

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135 146 159 167

8 Operator Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.1 8.2 8.3 8.4

Uniform Crossnorms . . Tensor Norms . . . . . . . Dual Norms . . . . . . . . . Additional Propositions

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179 185 192 199

9 Tensor Product Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 9.1 9.2 9.3 9.4

Tensor Product of Bounded A Hilbert-Space Setting . . . Some Spectral Properties . . Additional Propositions . . .

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203 209 217 228

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

1 Linear-Space Results

The chapter outlines essential algebraic notions required in the book. These are all related to linear spaces — in particular, to linear transformations and to quotient spaces of a linear space modulo a linear manifold. The purpose of such an introductory chapter is to put together only those results necessary in the forthcoming chapters. Algebraic aspects of bilinear maps will be discussed in Chapter 2. These first two chapters enable us to advance an axiomatic theory of algebraic tensor products of linear spaces in Chapter 3.

1.1 Notation, Terminology and Definitions This section summarizes the fundamental algebraic concepts, establishes the notation and terminology, and sets forth the pertinent definitions used in the text. The term algebraic concepts essentially means concepts related to linear space. The initial entries of this section focus on set-theoretic notions, mainly on the basic attributes of functions between abstract sets. The subsequent entries are all related to linear algebra: linear spaces and linear transformations. Image and range. The range of a function F : X → Y is the subset of the codomain Y of F consisting of all images F (x) for every x ∈ X, that is, range(F ) = {y ∈ Y : y = F (x) for some x ∈ X} = F (X). Injective, surjective and invertible. A function F : X → Y is surjective if F (X) = Y and injective (or one-to-one) if x1 = x2 in X implies F (x1 ) = F (x2 ) in Y , and it is invertible (or bijective) if it is injective and surjective. Extension and restriction. Let S be a subset of X. An extension of a function F : S → Y over a larger domain X is another function G : X → Y whose restriction to the smaller domain S coincides with the original function, that is, G|S = F, which means G(s) = F (s) for every s ∈ S, and so range(F ) = G(S). Composition. We use both notations HG or H ◦ G for composition of two functions H and G, say H(G(x)) for every x ∈ X where the sets X, Y, Z are © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Kubrusly, Bilinear Maps and Tensor Products in Operator Theory, Universitext, https://doi.org/10.1007/978-3-031-34093-2 1

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1. Linear-Space Results

such that G : X → Y and H : Y → Z. If F is a function of X into itself, then F n : X → X stands for the composition of F : X → X with itself n times. Idempotent and involution. A function F : X → X of a set X into itself is idempotent if F 2 = F (i.e., F (F (x)) = F (x) for every x ∈ X). Hence the range of an idempotent function F is precisely the set of all its fixed points. In fact, F = F 2 if and only if F (X) = {x ∈ X : F (x) = x}. Thus a function is idempotent if and only if it acts as the identity function I on its range. An involution is a function F : X → X such that F 2 = I.  Cartesian product. Let ∅ be the empty set. The Cartesian product γ∈Γ Xγ of an indexed family{Xγ }γ∈Γ of sets is the set consisting of all indexed families {xγ }γ∈Γ suchthat xγ ∈ Xγ for each γ ∈ Γ = ∅. In particular, if Xγ = X for all γ ∈ Γ, then γ∈Γ X = X Γ, where X Γ is the collection of all functions of Γ to X. More particularly, if In = {i ∈ N : i ≤ n} ⊆ N for some n ∈ N (N is the set Cartesian  product of n of positive integers), then write X n for X In so that the n copies of an arbitrary set X is written as X n = X In = i∈In X = i=1 X. The 2 Cartesian product of a pair of sets (X1 , X2 ) is denoted by X1 ×X2 = i=1 Xi . Throughout the book F denotes either the complex field C or the real field R . Let (X , +,  , F ) be a linear space, where X is an underlying set upon which

a binary operation (· + ·) : X ×X → X , called vector addition, and a function (·  ·) : F ×X → X , called scalar multiplication, are defined. In general, we refer to a linear space by writing simply X , and the scalar multiplication symbol  will always be omitted. The rest of this section is related to notation, terminology and definitions of basic linear-space concepts that will be required later. Linear manifold. A linear manifold M of a linear space X over F is a subset of X which is itself a linear space over F , with operations inherited from X . Linear manifolds are also called linear subspaces. Given a nonempty set S, if Y is a linear space over F , then so is the collection of all Y-valued functions on S, denoted by Y S. In particular, F S is the linear space of all F -valued functions on S. Let Lat(X ) denote the lattice of all linear manifolds of a linear space X . Hamel basis. Let S be a nonempty subset of a linear space X . The span of S, denoted by span S, is the smallest linear manifold of X including S (i.e., the intersection of all linear manifolds of X that include S), which coincides with the set of all (finite) linear combinations of vectors in S. If M = span S, then M is said to be spanned by the set S, and the set S is said to span the linear manifold M. A Hamel basis B for a linear space X is a linearly independent set spanning X (i.e., span B = X ). Every linear space has a Hamel basis, the cardinality of all Hamel bases for X is an invariant, and this constant cardinality is the dimension of X ; notation: dim X . If dim X = n for some n ∈ N 0 (N 0 is the set of nonnegative integers), then the linear space X is finite-dimensional ; otherwise it is infinite-dimensional (cf. Proposition 1.B in Section 1.4). Linear transformation. Let X and Y be linear spaces over the same field F . A linear transformation L : X → Y is an additive and homogeneous function.

1.1 Notation, Terminology and Definitions

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Let L[X , Y] be the linear space over F of all linear transformations from X to Y. If Y = X, write L[X ] = L[X , X ], the algebra of all linear transformations of X into itself. The range (or image space) and kernel (or null space) of L in L[X , Y] are denoted by R(L) = range(L) = L(X ) = {y ∈ Y : y = Lx for x ∈ X }, a linear manifold of Y, and N (L) = kernel(L) = L−1 ({0}) = {x ∈ X : Lx = 0}, a linear manifold of X , where {0} is the linear space containing the origin only. The null transformation is denoted by O : X → Y, and the identity transformation of X onto itself by I : X → X . The restriction L|M of a linear transformation L to a linear manifold M of X is a linear transformation in L[M, Y]. Linear functionals, algebraic dual and algebraic adjoint. For the particular case of Y = F the elements of L[X , F ] are referred to as linear functionals or linear forms, and the linear space L[X , F ] of all linear functionals on X is the algebraic dual of X , denoted by X  (i.e., X  = L[X , F ]). The value f (x) of f ∈ X  at x ∈ X is also denoted by x ; f . For every linear transformation L ∈ L[X , Y] consider the linear transformation L ∈ L[Y  , X  ] defined by L g = gL ∈ X  for every g ∈ Y  (i.e., (L g)(x) = g(Lx) ∈ F for every g ∈ Y  and every x ∈ X — this is also written as x ; L g = Lx ; g ). The linear transformation L ∈ L[Y  , X  ] is the algebraic adjoint of L ∈ L[X , Y]. Projections. A projection E : X → X is an idempotent linear transformation of a linear space X into itself. If E is a projection then so is I − E (this is the complementary projection of E), where I stands for the identity on X . Their range and kernel are related by R(E) = N (I − E) and N (E) = R(I − E). Isomorphisms. A linear transformation L ∈ L[X , Y] is injective (i.e., it is one-to-one) if and only if N (L) = {0}. By an isomorphism (or an algebraic isomorphism, or a linear-space isomorphism) we mean an invertible linear transformation between linear spaces. (The inverse of a linear transformation is linear, so the inverse of an isomorphism is an isomorphism.) We stick to the above terminology although it is not unique. Alternate jargons are: (a) linear transformation = homomorphism (or simply morphism), (b) linear transformation of X into itself = endomorphism, (c) injective linear transformation = monomorphism, (d) surjective linear transformation = epimorphism, (e) bijective linear transformation = isomorphism (same as above), (f) invertible linear transformation of X into itself = automorphism. Isomorphic linear spaces. Two linear spaces X and Y (over the same field) are isomorphic if there is an isomorphism of one onto the other. In this case (i.e., if X and Y are isomorphic) we write X ∼ = Y — isomorphism is an equivalence relation: reflexive, transitive, and symmetric. Two linear spaces are isomorphic if and only if they have the same dimension. Isomorphic equivalence — Isomorphic linear transformations. Let X , X, Y, Y be linear spaces over the same field, where X and X, as well as Y

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∼ Y). Two linear transformations  are isomorphic (i.e., X ∼ and Y, = X and Y =    ∼ L ∈ L[X , Y] and L ∈ L[X , Y] are isomorphically equivalent (notation: L = L)   if there exist isomorphisms J ∈ L[X , X ] and K ∈ L[Y, Y] such that  KL = LJ,  = K LJ −1 . This means the following diagram commutes: that is, L L

X −−−→ ⏐ ⏐ J

Y ⏐ ⏐ K

 L X −−−→ Y .

For the isomorphically equivalent case we get  ∼  ∼ N (L) = N (L) and R(L) = R(L).  = 0} (Indeed, with x  = Jx, J(N (L)) = {Jx ∈ X : Lx = 0} = { x ∈ X : LJ −1 x −1 −1   x = 0} = N (L)  and R(L)  = L(  X) x  = 0} = { x ∈ X : L = { x ∈ X : K LJJ  = L(J(X )) = KLJ −1 J(X ) = KL(X ) = KR(L).) Particular cases.   ∈ L[X] are similar. If X = Y and JL = LJ, then L ∈ L[X ] and L −1  then L ∼  ∈ L[X , Y].  If X = X and L = K L, = L for L ∈ L[X , Y] and L ∼      If Y = Y and L = LJ, then L = L for L ∈ L[X , Y] and L ∈ L[X , Y]. Embeddings. An embedding (or a linear-space embedding) is an isomorphism onto its range. A linear space X is embedded (or algebraically embedded ) in a linear space Y if there exists an embedding (i.e., an injective linear transformation) J : X → Y or, equivalently, if J : X → J(X ) ⊆ Y is an isomorphism between the linear space X and its range R(J) = J(X ). In this case J embeds X into Y. Notation: J : X → Y; if J either is clear in the context or is immaterial, then we write X → Y. (If X is a linear manifold of Y then the identity on X , viz., I : X → X ⊆ Y, trivially embeds X in Y.) Ordinary sum of linear manifolds. Let M and N be linear manifolds of a linear space X . The sum (or ordinary sum) of M and N is the linear space of X consisting of all sums of vectors from M with vectors from N :   M + N = x ∈ X : x = u + v for arbitrary u ∈ M and v ∈ N . The ordinary sum of an arbitrary family of linear manifolds is similarly defined. Algebraic complement. Let M be a linear manifold of a linear space X . An algebraic complement of M is any linear manifold N of X such that M + N = X and M ∩ N = {0}. Every linear manifold has an algebraic complement. Two linear manifolds M and N satisfying the above identities are referred to as algebraic complements or simply as complementary. Two linear manifolds M and N are algebraically disjoint if M ∩ N = {0}. Thus a pair

1.1 Notation, Terminology and Definitions

5

of algebraic complements is a pair of algebraically disjoint linear manifolds that sum to X . The dimension of any algebraic complement of M is another invariant for M, referred to as the codimension of M; notation codim M. The range and the kernel of a projection are complementary linear manifolds. In fact, M and N are complementary linear manifolds of a linear space X if and only if there exists a projection E ∈ L[X , X ] with R(E) = M and N (E) = N . Direct sum of linear spaces. Let X and Y be linear spaces over the same field F . The set consisting of all pairs (x, y) with x ∈ X and y ∈ Y is the Cartesian product X ×Y of X and Y. This is not a linear space by itself. However, it may be equipped with a vector addition ⊕ called direct sum and a scalar multiplication that make it into a linear space (X ×Y, ⊕,  , F ) under the common field F . This linear space, whose underlying set is the Cartesian product X ×Y of the linear spaces X and Y, is denoted by X ⊕ Y = (X ×Y, ⊕,  , F ) and is referred to as the direct sum, or the full algebraic direct sum, or the external direct sum of X and Y. The operations of addition (direct sum) and scalar multiplication are defined by (x1 , y1 ) ⊕ (x2 , y2 ) = (x1 + x2 , y1 + y2 )

and

α(x, y) = (αx, αy)

for every (x, y), (x1 , y1 ), (x2 , y2 ) in X ⊕ Y and every α in F . In other words, the direct sum X ⊕ Y of a pair of linear spaces X and Y is a linear space consisting of all pairs (x, y) in the Cartesian product X ×Y when vector addition and scalar multiplication are  defined as above. A direct sum of an arbitrary family having the Cartesian of linear spaces (e.g., γ∈Γ Xγ ) is defined accordingly,  product of such a family of linear spaces (e.g., γ∈Γ Xγ ) as the underlying set.

n n In particular, for a linear space X over F let X n = i=1 X = i=1 X , ⊕,  , F be the linear space consisting of the direct sum of n copies of X . Equivalently,   write F n (X ) = X In = (x1 , · · · , xn ) : xi ∈ X , i ∈ In = {1, . . . , n} for the collection of all n-tuples of vectors in X , equipped with the direct sum algebra. Thus these are different notations for the same linear space, and so we write n X n = F n (X ) = X. i=1  n In particular, F n = F n (F ) = i=1 F . Another important particular case refers to the direct sum of a pair of linear manifolds of the same linear space (sometimes called internal direct sum). If M and N are linear manifolds of a linear space X , then the ordinary sum M + N is isomorphic to the (internal) direct sum M ⊕ N if and only if M and N are algebraically disjoint,   M+N ∼ = M ⊕ N ⇐⇒ M ∩ N = {0}. Thus we identify ordinary and direct sums of algebraic complements. Again, direct sums of more than a pair of linear manifolds are similarly defined. Formal linear combination. Let S be an arbitrary set and let F be a field. Suppose S = ∅ to avoid trivialities. Consider the linear space F S of all functions from S to F (i.e., of all scalar-valued functions f : S → F on S),

6

1. Linear-Space Results

where vector addition and scalar multiplication are pointwise defined (see, e.g., [52, Example 2.E]): for every f, g ∈ F S and every α ∈ F , (f + g)(s) = f (s) + g(s)

and

(αf )(s) = αf(s)

for every s ∈ S.

Let # stand for cardinality and consider the set   S / = f ∈ F S : f (S\A) = 0 for some A ⊆ S with #A < ∞ of all functions f : S → F that vanish everywhere on the complement of some finite subset A of S (which depends on f ). This S / is a linear manifold of F S. / . Take an arbitrary (Indeed, αf clearly lies in S / for every α ∈ F and every f ∈ S pair f, g ∈ S / and let Af and Ag be finite subsets of S — so that Af ∪ Ag ⊆ S is finite — for which f (S\Af ) = g(S\Ag ) = 0. If s ∈ Af ∪ Ag , then f (s) = g(s) = 0 and so (f + g)(s) = f (s) + g(s) = 0, which implies f + g ∈ S / . There/ fore S / is itself a linear space over F , whose origin is the zero function 0 ∈ S — defined by 0(s) = 0 ∈ F for all s ∈ S.) Now for each s ∈ S take the characteristic function es = χ{s} : S → F of the singleton {s} ⊆ S (i.e., es (t) = 1 ∈ F if t = s ∈ S and zero otherwise). /. The set S = {es }s∈S is a Hamel basis for the linear space S In fact, each function es lies in S / since es (S\{s}) = 0 for each s ∈ S. Moreover, if s, t ∈ S and t = s, then es (t) = et (s) = 0, so that {es }s∈S is linearly independent. Also, since an arbitrary vector f ∈ S / is a scalar-valued function that takes nonzero values only over a finite subset of S, say on {si }ni=1 , then n αi esi ∈ S /, f= i=1

with αi ∈ F . So the vectors f ∈ S / are (finite) linear combinations of vectors in / . In other words, span {es }s∈S = S / . Thus S = {es }s∈S ⊂ S / ⊆ FS {es }s∈S ⊂ S is a Hamel basis for S / . This linear space S / is referred to as the free linear / = #S = #S. This space generated by S. Since #{es }s∈S = #S, we get dim S sets up a natural identification ≈ such that s ≈ es and so S ≈ S , which in turn leads to a natural identification for an arbitrary linear combination in S /, n n αi esi ≈ αi si , i=1 i=1

n where i=1 αi si is referred to as a formal linear combination of points si ∈ S (although addition or scalar multiplication is not directly defined on the set S), the collection of which is the linear space of formal linear combinations from S. So any function f in the linear space S / is identified with a formal linear combination of points in S, and the set S that generates the free linear / . In this sense the set S may space S / is identified with the Hamel basis S for S be regarded as a subset of S / , and a function f in S / may be regarded as a formal linear combination. Thus write n n αi si for αi si ≈ f ∈ S / and S⊂S / for S ≈ S ⊂ S /. f= i=1

i=1

1.2 Extension of Linear Transformations

7

Summing up: Every set S generates a linear space S / over F , which is a F S, where every vector in S / is identified linear manifold of the linear space n with a formal linear combination i=1 αi si of elements si of S.

1.2 Extension of Linear Transformations Extension of linear transformations will play a crucial role throughout the book. An algebraic extension result for linear transformations goes as follows. Theorem 1.1. Let X and Y be linear spaces over the same field F and let M be a linear manifold of X . If L : M → Y is a linear transformation, then there  : X → Y of L defined on the whole space X . exists a linear extension L Proof. We borrow the proof from [52, Theorem 2.9]. Let Lat(X ) stand for the lattice of all linear manifolds of a linear space X . Take L ∈ L[M, Y] and set    ∈ L[N , Y] : N ∈ Lat(X ), M ⊆ N and L = L| M , L= L the collection of all linear transformations from linear manifolds of X to Y which are extensions of L. This L is nonempty since at least L ∈ L — it will be clear below that the set L contains  plenty of elements. Also, since L is a subcollection of the collection F = S⊆ X Y S of all functions into Y whose domain S is included in X , the set L is partially ordered in the extension   γ } in L has a supremum  L ordering ≤, and every chain {L γ γ in F with domain 



 γ =  D(L  γ =  R(L  γ ) and range R  L  γ ) (see, e.g., [52, Problem D L γ

γ

γ

γ

 γ ) ∈ Lat(X ), because each L  γ is a linear transformation 1.17]). Since D(L defined on a linear manifold of X , and since Lat(X ) is a complete lattice (i.e., 

γ = every subset of it has a supremum and an infimum), the domain D γ L

   γ =  γ ) lies in Lat(X ). Note as well that R  γ ) ∈ Lat(Y). L D(L R(L γ

  Claim. The supremum γ L γ lies in L.

γ

γ



 γ =  D(L  γ ) ∈ Lat(X ). Proof of Claim. Take arbitrary vectors u, v ∈ D γ L γ  λ ) for some L  λ ∈ {L  γ } and v ∈ D(L  μ ) for some L  μ ∈ {L  γ }. As Thus u ∈ D(L      {Lγ } is a chain, we get Lλ ≤ Lμ (or vice versa), and so D(Lλ ) ⊆ D(Lμ ). Then  μ (αu + βv) = αL μ u + β L  μ v for every α, β ∈ F  μ ) and hence L αu + βv ∈ D(L



    γ (αu + βv) =   (since each Lγ is linear). Then γ Lγ |D(Lµ ) = Lμ . Thus γ L 





  γ u + β  L    α γL γ γ v, and so γ Lγ : D γLγ → Y is linear. Also, Lγ |M = L

   γ } is a chain, so that  γ |M = L. Conclusion:  γ ∈ L. and {L L L   γ

γ

Then every chain in L has a supremum (hence an upper bound) in L. Thus  0 : N0 → Y. Zorn’s Lemma ensures that L contains a maximal element, say L  0 is a linear extension of L The program is to show that N0 = X , and so L over X . A proof by contradiction goes as follows. Suppose N0 = X . Take x1 ∈ X \N0 (and so x1 = 0 because N0 is a linear manifold of X ) and consider the sum of N0 and the one-dimensional linear manifold of X spanned by {x1 },

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N1 = N0 + span {x1 }. This is a linear manifold of X properly including M (as M ⊆ N0 ⊂ N1 ). Since N0 ∩ span {x1 } = {0}, every x in N1 has a unique representation as a sum of a vector in N0 and a vector in span {x1 }. Thus for each x ∈ N1 there is a unique pair (x0 , α) in N0 ×F for which x = x0 + αx1 . (In fact, if x = x0 + αx1 = x0 + α x1 , then x0 − x0 = (α − α)x1 lies in N0 ∩ span {x1 } = {0} so that x0 = x0 and α = α — since x1 = 0.)  1 : N1 → Y defined for each x ∈ N1 by Take any y ∈ Y and consider the map L 1 x = L  0 x0 + αy, L  0 ) and L  1 |N = L  0 (and so L 0 ≤ L  1 ). which is linear (inherits linearity from L 0    Since M ⊆ N0 ⊂ N1 , we get L = L0 |M = L1 |M . Thus L1 ∈ L, which contra 0 = L  1 ). Hence N0 = X .  0 is maximal in L (for L   dicts the fact that L Remark 1.2. Two Basic Properties of Linear Extension. (a) Linear extension is not unique. For instance, with {0}= M ⊂ X = Y, the identity transformation I : M → M ⊂ X acts as the natural embedding of M into X , and any projection E ∈ L[X , X ] with R(E) = M linearly extends I over X . Thus take arbitrary linear spaces X , Y over F , an arbitrary (but fixed) linear manifold M of X , and consider a relation ∼ (or ∼M ) on L[X , Y] given for L, L ∈ L[X , Y] by L ∼ L ⇐⇒ L |M = L|M . This ∼ is an equivalence relation (reflexive, transitive and symmetric). For each L ∈ L[X , Y] take the equivalence class [L] = {L ∈ L[X , Y] : L ∼ L} = {L ∈ L[X , Y] : L |M = L|M }, and consider the quotient space L[X , Y] / ∼ = L[X , Y] /M = {[L] ⊆ L[X , Y] : L ∈ L[X , Y]} consisting of all equivalence classes [L] for every L ∈ L[X , Y]. Such an equivalence relation ∼ is linear, which means that for every L1 , L1, L2 , L2 ∈ L[X , Y] and every α ∈ F , L1 ∼ L1 and L2 ∼ L2

imply

L1 + L2 ∼ L1 + L2 and αL1 ∼ αL1

(i.e., if L1 |M = L1 |M and L2 |M = L2 |M , then (L1 + L2 )|M = L1 |M + L2 |M = L1 |M + L2 |M = (L1 + L2 )|M , and so L1 + L2 ∼ L1 + L2 — also, αL1 ∼ αL1 ). Thus (see, e.g., [52, Example 2G]) addition and scaling are defined by [L1 ] + [L2 ] = [L1 + L2 ]

and

α[L1 ] = [αL1 ]

in the linear space L[X , Y] /M over F (this sort of linear space is the subject of the next section). Take the extension operation  : L[M, Y] → L[X , Y] /M between linear spaces over the same field given by  = [L]  L

for every

L ∈ L[M, Y],

1.3 Quotient Space

9

 ∈ L[X ,Y] on the right-hand side (inside square brackets) means any where L individual extension of L ∈ L[M, Y] as in Theorem 1.1 and, using the same  ∈ L[X ,Y] /M on the left-hand side (i.e., L  = [L]  ∈ L[X ,Y] /M) notation, L is the value of the extension operation  at a given L ∈ L[M, Y]. (b) The extension operation  is linear. Indeed, for arbitrary L1 , L2 in L[M, Y] and for arbitrary individual extensions of them (same notation for both), 1 , L 2 in L[X , Y], we get (L1 1 |M + L 2 |M = say L + L2 )|M = L1 + L2 = L      (L1 + L2 )|M , and [L1 + L2 ] = [L1 + L2 ] so that (see argument in item (a)) 1 + L 2 ] = [L 1 ] + [L 2 ] = L 1 + L 2 . L1 + L2 = [L1 + L2 ] = [L  = αL  (i.e., [α L]  = α[L])  for every L ∈ L[M, Y] and α ∈ F . Similarly, αL

1.3 Quotient Space Let M be a linear manifold of a linear space X over a field F , and consider a relation ∼ on X defined as follows. If x and x are vectors in X , then x ∼ x ⇐⇒ x − x ∈ M. That is, x ∼ x if x is congruent to x modulo M — notation: x ≡ x(mod M). Since M is a linear manifold of X , the relation ∼ in fact is an equivalence relation on X (reason: 0 ∈ M — reflexivity, x − x = (x − x) + (x − x ) ∈ M whenever x − x and x − x lie in M — transitivity, and x − x ∈ M whenever x − x ∈ M — symmetry). The equivalence class (with respect to ∼)     [x] = x ∈ X : x ∼ x = x ∈ X : x = x + z for some z ∈ M = x + M of a vector x in X is called the coset of x modulo M. The quotient space X /M of X modulo M is the collection of all cosets [x] modulo M for every x ∈ X . In other words, X /M is precisely the collection of all equivalence classes [x] with respect to the equivalence relation ∼ for every x in X . Equivalently, the quotient space X /M of X modulo M is the collection of all translations of the linear manifold M (also called affine spaces or linear varieties), and so for each vector x ∈ X the coset [x] = x + M ∈ X/M is the translation of M by x. The equivalence relation ∼ is a linear equivalence relation on the linear space X in the sense that if x ∼ x and y  ∼ y, then x + y  ∼ x + y and αx ∼ αx. Indeed, if x − x ∈ M and y  − y ∈ M, then (x + y  ) − (x + y) = (x − x) + (y  − y) ∈ M and αx − αx = α(x − x) ∈ M for every scalar α in F . Therefore, with vector addition and scalar multiplication defined by [x] + [y] = [x + y]

and

α[x] = [αx],

the quotient space X /M of the linear space X modulo a linear manifold M is made into a linear space over the same scalar field. The origin [0] of X /M is

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1. Linear-Space Results

[0] = M ∈ X /M

(i.e., [y] = [0] ⇐⇒ y ∈ M).

So, as [x] = x + M for x ∈ X , the trivial cases are (∼ = denotes isomorphic to): M = {0} ⇐⇒ [0] = {0} ⇐⇒ [x] = {x} ∀ x ∈ X ⇐⇒ X /M = X /{0} ∼ =X, M = X ⇐⇒ [0] = X ⇐⇒ [x] = [0] ∀ x ∈ X ⇐⇒ X /M = X /X = {[0]} ∼ = {0}. The quotient map (or the natural quotient map) π : X → X /M of the linear space X onto the linear space X /M is defined by π(x) = [x] = x + M for every x ∈ X , which is a surjective linear transformation according to the definition of addition and scalar multiplication in X /M. Indeed, for every x ∈ X and α ∈ F π(x+y) = [x+y] = [x]+[y] = π(x)+π(y) and π(αx) = [αx] = α[x] = απ(x). The kernel N (π) and range R(π) of the quotient map π are given by   N (π) = π −1 ([0]) = x ∈ X : π(x) = [0]     = x ∈ X : [x] = [0] = x ∈ X : x ∈ M = M ⊆ X ,   R(π) = π(X ) = [y] ∈ X /M : [y] = π(x) for some x ∈ X   = [y] ∈ X /M : y + M = x + M for some x ∈ X = X /M. So X /M = X /N (π) = R(π) (which actually is a property that, up to an isomorphism, holds for every linear transformation — cf. Theorem 1.4(b) below), and the quotient map π ∈ L[X , X /M] is a linear transformation such that (i)

π is surjective,

(ii) π is injective if and only if M = {0}, (iii) π = 0 if and only if M = X . Remark 1.3. Linear Manifolds and Restrictions. The notions of subsets and linear manifolds of a quotient space and of restrictions of functions on quotient spaces require some attention, and it is advisable to refine the notation. (a) Write [x]M = x + M ∈ X /M for the coset of x ∈ X modulo a linear manifold M ⊆ X of the linear space X . An element of X /M is of the form x + M for a fixed linear manifold M of X and for an arbitrary x ∈ X , and therefore a subset V ⊆ X /M of the quotient space X /M consists of elements s + M for arbitrary vectors s ∈ S in an arbitrary subset S ⊆ X of the linear space X : V is a subset of X /M ⇐⇒ V = S/M for a subset S of X . Thus, as is readily verified, V is a linear manifold of X /M ⇐⇒ V = S/M for a linear manifold S of X .

1.3 Quotient Space

11

Therefore, if S is a linear manifold of X , then S/M is a linear manifold of X /M made up of all cosets [s]M = s + M such that s ∈ S ⊆ X . Thus S ⊆ X =⇒ S/M ⊆ X /M. (b) Consider the case of linear manifolds of M rather than linear manifolds of X . Fix an arbitrary x ∈ X and let R be an arbitrary linear manifold of the linear manifold M, and consider the coset [x]R = x + R ∈ X /R as an element of the quotient space X /R. Since R ⊆ M, we get [x]R = x + R ⊆ x + M = [x]M . However, the coset [x]R = x + R ∈ X /R is not an element of the quotient space X /M which consists of elements of the form x + M only, for each x ∈ X and for a fixed M ⊆ X . In other words, although R ⊆ M =⇒ x + R ⊆ x + M for each x ∈ X , the inclusion of quotient spaces fails: R ⊆ M =⇒ / X /R ⊆ X /M. Actually, every [x]R ∈ X /R does lie in X /M. (c) In particular, for R ⊆ M ⊆ X , R/{0} ⊆ M/{0} ⊆ X /{0}, although, for {0} ⊂ R ⊂ M ⊂ X , X /{0} ⊆ / X /R ⊆ / X /M. (d) We may naturally identify each x ∈ X with [x]{0} = x + {0} ∈ X /{0} ∼ = X, and so we may regard each x ∈ X as an element of X /{0}. Analogously, we may identify X with X /{0}. However, if M = {0}, then X /{0} ⊆ / X /M, and we may not regard each x in X when identified with [x]{0} as an element of X /M, as we may not regard X /{0} as a linear manifold of X /M. (e) Now take a function F : X /M → Z on the quotient space X /M. It is clear that restrictions F |V of it only make sense if over a subset V of its domain X /M. As we saw in (a), this holds only if V = S/M for some subset S of X . Theorem 1.4. Let X and Z be linear spaces over the same field, let M be a linear manifold of X , consider the quotient space X /M, and take the natural quotient map π : X → X /M. (a) (Universal Property): If L ∈ L[X , Z] and M ⊆ N (L), then there exists  ∈ L[X /M, Z] such that a unique L  ◦ π. L=L In other words, the diagram L

X −−−→ 

π



Z  ⏐ ⏐L X /M

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commutes, which means the quotient map π factors the linear transforma = N (L)/M and R(L)  = R(L). tion L through X /M. Moreover, N (L) (b) (First Isomorphism Theorem): R(L) ∼ = X /N (L) for every L ∈ L[X , Z]. Proof. (a) Let X and Z be linear spaces over the same field, let M be a linear manifold of X , and consider the quotient space X /M. Let L : X → Z be a linear transformation, and consider the natural quotient map π : X → X /M, which is given by π(x) = [x] for every x ∈ X . Claim. If M ⊆ N (L), then there exists a unique linear transformation  : X /M → Z for which L = L  ◦ π. L  : X /M → Z by Proof of Claim. Define a transformation L  = Lx L[x]

for every

[x] ∈ X /M.

As a function, such a definition makes sense if and only if  1 ] = L[x  2 ] =⇒ [x1 ] = [x2 ], [x1 ], [x2 ] ∈ X /M and L[x  1] = which means if [x1 − x2 ] = [0] for arbitrary [x1 ], [x2 ] ∈ X /M, then L[x  2 ]. Since L is linear and since [0] = M, this is equivalent to L[x x1 , x2 ∈ X and x1 − x2 ∈ M =⇒ L(x1 − x2 ) = 0, which in turn (as M is a linear space) is equivalent to Lx = 0 for every x ∈ M,  which is equivalent to that is, M ⊆ N (L). Thus the definition of L,  ◦ π = L, L  is well-defined) if and only if makes sense (i.e., L M ⊆ N (L).  is linear since linearity of L is readily transferred to L,  and Moreover, L uniqueness follows at once.  1 ] + [x2 ]) = L[x  1 + x2 ] = L(x1 + x2 ) = Lx1 + Lx2 = L[x  1 ] + L[x  1] (Indeed, L([x    and L(α[x]) = L[αx] = L(αx) = αLx = α L[x], ensuring linearity, and if L =  ◦ π = L  ◦ π, then L   [x] = L[x]  and so L   = L,  ensuring uniqueness.) L   Also, since π : X → X /M is surjective (i.e., π(X ) = X /M), we get  = L(X  /M) = L(π(X   ◦ π)(X ) = L(X ) = R(L), R(L) )) = (L  = L(π(x))   ◦ π)(x) = Lx for every [x] ∈ X /M, so that and locally L[x] = (L  = 0 if and only if x ∈ N (L), which implies L[x]      = [x] ∈ X /M : L[x]  = 0 = [x] ∈ X /M : x ∈ N (L) = N (L)/M. N (L)

1.3 Quotient Space

13

 = M/M = M = [0], which (b) In particular, for M = N (L) we get N (L)   ⊆ Z is injective, and so means the linear transformation L : X /N (L) → R(L)   = R(L). it is an isomorphism of X /N (L) onto R(L). But R(L)   If S is an arbitrary subset of X , then   π(S) = [y] ∈ X /M : [y] = π(s) for some s ∈ S   = [y] ∈ X /M : [y] = [s] for some s ∈ S   = [y] ∈ X /M : [y] − [s] = [0] for some s ∈ S   = [y] ∈ X /M : [y − s] = [0] for some s ∈ S   = [y] ∈ X /M : y − s ∈ M for some s ∈ S   = [y] ∈ X /M : y ∈ s + M for some s ∈ S   = [y] ∈ X /M : y ∈ S + M . In particular, π(X ) = {[y] ∈ X /M : y ∈ X } = X /M since X + M = X (i.e., π is surjective). Hence the inverse image of π(S) under π is     π −1 (π(S)) = x ∈ X : π(x) ∈ π(S) = x ∈ X : [x] ∈ π(S)   = x ∈ X : x + M ∈ M + S = S + M. If N is a linear manifold of X for which X = M + N , then     π(N ) = [y] ∈ X /M : y ∈ N + M = [y] ∈ X /M : y ∈ X = X /M, and in this case the restriction π|N : N → X /M of π to N is surjective as well. Moreover, if M ∩ N = ∅, then   −1 N (π|N ) = π|N ([0]) = v ∈ N : π(v) = [0]    = v ∈ N : [v] = [0] = v ∈ N : v ∈ M} = {0}, and in this case the restriction π|N : N → X /M of π to N is injective. Since π is linear, its restriction to a linear manifold is again linear. Thus if N is an algebraic complement of M, then π|N : N → X /M is an algebraic isomorphism (referred to as the natural quotient isomorphism) between the linear spaces N and X /M: every algebraic complement of M is isomorphic to the quotient space X /M. Again, with ∼ = standing for isomorphism, N ∼ = X /M for every algebraic complement N of M. Then every algebraic complement of M has the same (constant) dimension: dim N = dim X /M for every algebraic complement N of M, and so two algebraic complements of M are algebraically isomorphic (cf. Section 1.1). Thus (see, e.g., [52, Lemma 2.17, Theorem 2.18]). codim M = dim X /M.

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1. Linear-Space Results

Remark 1.5. Hamel Basis. Let M, N , and R be linear manifolds of a linear space X . A set {[eδ ]}δ∈Δ of cosets modulo M (with each eδ in X ) is a Hamel basis for the linear space X /M if and only if {eδ }δ∈Δ is a Hamel basis for some algebraic complement of M (see, e.g., [56, Remark A.1(b)]). Hence every Hamel basis BX = {eγ }γ∈Γ for X is such that {[eγ ]}γ∈Γ includes a Hamel basis BX /M for X /M. Since the quotient map π : X → X /M is surjective, the image π(BX ) of an arbitrary Hamel basis BX for X includes a Hamel basis BX /M for X /M (i.e., BX /M ⊆ π(BX )). Therefore span π(BX ) = X /M

for every Hamel basis BX for X ,

and so every element [x] of X /M can be written as a (finite) linear combination of images under π of elements of an arbitrary Hamel basis for X . (This result plays an important role in Section 4.1 in the forthcoming Chapter 4.)

1.4 Additional Propositions As before, X , Y, Z denote nonzero linear spaces over the same field F , M and N stand for linear manifolds of a fixed linear space, L[X , Y] denotes the linear space over F of all linear transformations of X into Y, and X  = L[X , F ] denotes the algebraic dual of X . Also, as everywhere in this chapter, ∼ = stands for (algebraic) isomorphism. A subset S of a linear space is linearly independent if each vector s in S is not a (finite) linear combination of vectors in S\{s}. Proposition 1.A. Take any ∅ = S ⊂ X . The assertions below are equivalent. (a) S is linearly independent. (b) Each nonzero vector in span S has a unique representation as a linear combination of vectors in S. (c) Every finite subset of S is linearly independent. (d) There is no proper subset of S whose span coincides with span S. Proposition 1.B. Every linear space X has a Hamel basis, the empty set ∅ is the Hamel basis for the trivial space X = {0} (note that {0} is not linearly independent), and every Hamel basis for a linear space has the same cardinality. (Such an invariant — i.e., this constant cardinality — is the dimension of X , denoted by dim X , and so dim X = 0 if and only if X = {0}.) Proposition 1.C. Two linear spaces over the same field are isomorphic if and only if they have the same dimension. Proposition 1.D. L ∈ L[X , Y] is injective if and only if N (L) = {0}. Proposition 1.E. This is the rank and nullity identity. If L ∈ L[X , Y], then dim N (L) + dim R(L) = dim X .

1.4 Additional Propositions

15

Proposition 1.F. f ∈ X  is injective if and only if X is isomorphic to F . Proposition 1.G. If x ∈ X ∩ N (f ) for all f ∈ X  , then x = 0. Three equivalent ways of stating Proposition 1.G go as follows. (i) If x ∈ X is such that f (x) = 0 for all f ∈ X  , then x = 0. (ii) If x ∈ X is nonzero, then there exists f ∈ X  for which f (x) = 0. (iii) If A ⊆ X is such that f (A) = 0 for all f ∈ X  , then A = {0}. Clearly, all converses trivially hold true. Proposition 1.H. If 0 = f ∈ X  and x ∈ N (f ), then the one-dimensional linear manifold span {x} and N (f ) are algebraic complements: span {x} ∩ N (f ) = {0}

and

span {x} + N (f ) = X .

Proposition 1.I. If N (f ) = N (g) for f, g ∈ X  , then f = αg for 0 = α ∈ F . Proposition 1.J. Let F stand for either R or C , let # be the collection of all F -valued sequences with a finite number of nonzero entries, and let P[0, 1] be the collection of all polynomials in the variable t ∈ [0, 1] with coefficients in F of any degree. Then P[0, 1] and # are isomorphic linear spaces over F . Proposition 1.K. Every linear manifold has an algebraic complement. Proposition 1.L. Every algebraic complement of a linear manifold M of a linear space X is isomorphic to the quotient space X /M. Proposition 1.M. The dimension of any algebraic complement of a linear manifold is invariant (called codimension of M and denoted by codim M). Proposition 1.N. Two linear manifolds M and N of a linear space X are algebraic complements of each other if and only if there exists a projection E ∈ L[X , X ] such that R(E) = M and N (E) = N . Proposition 1.O. If M and R are linear manifolds of a linear space X , then M + R = X and dim R < ∞

=⇒

codim M < ∞

even if R is not algebraically disjoint from M. Proposition 1.P. If M and N are algebraic complements in X , then dim X = dim M + dim N . Thus if M is a linear manifold of a linear space X , then dim X = dim M + dim X /M = dim M + codim M according to Propositions 1.C, 1.L, 1.M and 1.P.

16

1. Linear-Space Results

Proposition 1.Q. Let M and N be linear manifolds of a linear space X . The following assertions are pairwise equivalent. (a) M ∩ N = {0}

(i.e., M and N are algebraically disjoint).

(b) For each x in M + N there exists a unique u in M and a unique v in N such that x = u + v. (c) The natural mapping Φ : M ⊕ N → M + N (which is given by Φ(u, v) = u + v for every u ∈ M and v ∈ N ) is an isomorphism. Proposition 1.R. Let M and N be linear manifolds of a linear space. (a) If M and N are algebraically disjoint, then dim(M ⊕ N ) = dim(M + N ) = dim M + dim N . (b) If M and N are finite-dimensional, then dim(M + N ) = dim M + dim N − dim(M ∩ N ). Proposition 1.S. Let X be a linear space and consider its decomposition X = M⊕N as a direct sum of algebraically disjoint linear manifolds M and N of X . Let E in L[X , X ] be the projection on M along N (i.e., R(E) = M and N (E) = N ) and let P = I − E be the complementary projection on N along M. With respect to the decomposition X = M ⊕ N every linear transformation L in L[X , X ] can be written as a 2×2 matrix with linear transformation entries   A B L = , C D where A, B, C, D are linear transformations given by A = EL|M : M → M, B = EL|N : N → M, C = P L|M : M → N , and D = P L|N : N → N . The linear transformation L ∈ L[X , X ] is said to be the direct sum of the linear transformations A ∈ L[M, M] and D ∈ L[N , N ], written as L=A⊕D if B and C are null (i.e., B = O ∈ L[N , M] and C = O ∈ L[M, N ]). Moreover, with respect to the same decomposition the projections E and P are written as (where I and O stand for the identity and null transformations) E =I ⊕O

and

P = O ⊕ I.

Proposition 1.T. If {Mγ } is an indexed family of pairwise algebraically disjoint  linear manifolds of a linear space, say X , then the dimension of the direct sum γ Mγ is the sum of the dimensions of each Mγ .

1.4 Additional Propositions

17

Proposition 1.U. Let M and N be linear manifolds of a linear space X , and let L : X → Z be a linear transformation of X into any linear space Z. (a) R(L) ∼ = X /N (L) (this is the First Isomorphism Theorem — as in Theorem 1.4). (b) (M + N )/N ∼ = M/(M ∩ N ) (this is referred to as the Second Isomorphism Theorem). (c) If M ⊆ N , then (X /M)/(N /M) ∼ = X /N (this is referred to as the Third Isomorphism Theorem). (d) If M and N are algebraic complements and M and N  are linear manifolds of M and N , then (M ⊕ N )/(M ⊕ N  ) ∼ = M/M ⊕ N /N  (where the direct sums on the left-hand side are internal direct sums, and the direct sum on the right-hand side is an external direct sum). Proposition 1.V. Let X  be the algebraic dual of a linear space X . Then dim X ≤ dim X  ,

and

dim X = dim X 

⇐⇒

dim X < ∞.

Notes: The above propositions summarize some of the results discussed in the main body of this chapter, and introduce some results that will be required in the upcoming chapters. These are all standard results whose proofs can be found in essentially every text dealing with linear space. The books listed below are among the many possible sources. In particular, for Propositions 1.A to 1.E see, e.g., [52, Proposition 2.3, Theorems 2.5 to 2.8, 2.12, Problem 2.17]. In addition to being well-known useful results, Propositions 1.F to 1.I will often be used in the remaining portions of the present book, and so their proofs are sketched below. Proof of Proposition 1.F. By Propositions 1.D and 1.E, if f is injective, then dim R(f ) = dim X . Since X = {0} and R(f ) ⊆ F , it follows that 0 < dim X = dim R(f ) ≤ dim F = 1. Thus dim X = 1, which means X is isomorphic to F by Proposition 1.C. The converse is trivially verified.   Proof

of Proposition 1.G. Let {eγ } be a Hamel basis for X . Take any x ∈ X . Let x = γ αγ (x)eγ be its unique (finite) expansion in terms of {eγ }. Consider the functional αγ : X → F associating to each x the coefficient αγ (x) in its expansion. As is readily verified, it is linear. So αγ ∈ X . If f (x) = 0 for all f ∈ X , then   αγ (x) = 0 for all γ and x = γ αγ (x)eγ = 0. The converse is trivial. Proof of Proposition 1.H. If f = 0, take x ∈ X \N (f ). If y ∈ span {x} ∩ N (f ), then y = αx for α ∈ F and 0 = f (y) = αf (x). Thus α = 0 (as x ∈ N (f )) so y = 0. Hence span {x} ∩ N (f ) ⊆ {0}. Now take y ∈ X . Since f (x) = 0, write (y) (y) (y) (y) x + (y − ff (x) x), where ff (x) x ∈ span {x} and f (y − ff (x) x) = 0 (i.e., y = ff (x) y=

f (y) f (x) x

∈ N (f )). Then y ∈ span {x} + N (f ). So X ⊆ span {x} + N (f ).

 

18

1. Linear-Space Results

Proof of Proposition 1.I. Let N (f ) = N (g). Set N = N (f ) = N (g). To avoid trivialities, N = X . Take any x ∈ X \N . By Proposition 1.H span {x} ∩ N = {0} and span {x} + N = X . Take an arbitrary y ∈ X . Write y = γ x + u for γ in F and u in N . Thus f (y) = γ f (x) and g(y) = γ g(x) for every y ∈ X . Since (y) g(y) (x) = γ = g(x) . Then set 0 = α = fg(x) ∈ F so that f (y) = αg(y).   x ∈ N , ff (x) Also see [81, Theorem 3.10] for Propositions 1.F to 1.I. For Propositions 1.J to 1.N see, e.g., [52, Example 2.M, Theorems 2.16 to 2.20], and for Proposition 1.O, [56, Remark A.1(a)]. Proposition 1.P is a straightforward consequence of the rank and nullity identity of Proposition 1.E with a little help from Proposition 1.N (also see [91, Theorem 6.2]). For Propositions 1.Q to 1.S see, e.g., [52, Theorem 2.14, Problem 2.25, Proposition 2.22]. Proposition 1.T says that   Mα ∩ Mβ = {0} ∀ α = β =⇒ dim Mγ = dim Mγ γ

γ

(see, e.g., [5, Problem

2.F]). Recall  from Section 1.1 that if Mα ∩ Mβ = {0} for all α = β, then γ Mγ ∼ = γ Mγ . Thus, in this case, we also get dim

 γ

 Mγ = dim Mγ γ

by Proposition 1.C. For further issues on direct sum as defined here see, e.g., [5, Chapter 2] and [52, Section 2.8]). For Proposition 1.U see, e.g., [81, Theorems 3.5, 3.7, 3.8, 3.9]. For Proposition 1.V see, e.g., [81, Theorem 3.12]. We close this first chapter with a remark on notation. As usual, ⊂ means proper inclusion — A ⊂ B means that the set A is a proper subset of another set B (A is included in B but is not equal to B), and A ⊆ B means that a set A is either a proper subset of a set B or is equal to B (i.e., ⊆ means ⊂ or =). Equality between two sets A and B means A ⊆ B and B ⊆ A. Thus ⊆ simply means that a set is not subset of another set (it is not included — not properly included and not equal), and hence A ⊂ B means A ⊆ B and B ⊆ A. The symbol  is somewhat similar to ⊂, it means an inclusion that may in certain cases be a proper inclusion.

Suggested Readings Berberian [3] Brown and Pearcy [5] Halmos [30, 32] Kubrusly [52, 56]

Lang [66] MacLane and Birkhoff [69] Roman [81] Taylor and Lay [91]

2 Bilinear Maps: Algebraic Aspects

Let X , Y, Z be nonzero linear spaces over a field F and consider the Cartesian product X ×Y (without imposing any algebraic structure on X ×Y besides the fact that both X and Y are linear spaces). A bilinear map φ : X ×Y → Z is a function from the Cartesian product of linear spaces to a linear space whose sections are linear transformations. Precisely, for each y ∈ Y let φy = φ(·, y) = φ|X ×{y} : X → Z be the y-section of φ, and for each x ∈ X let φx = φ(x, ·) = φ|{x}×Y : Y → Z be the x-section of φ. The map φ is bilinear if φy and φx are linear transformations, that is, if φy = φ(·, y) ∈ L[X , Z] and φx = φ(x, ·, ) ∈ L[Y, Z]. In other words, φ is bilinear if for every x, x1 , x2 ∈ X , y, y1 , y2 ∈ Y, and α, α1 , α2 , β1 , β2 ∈ F , φ(x1 + x2 , y) = φ(x1 , y) + φ(x2 , y),

φ(x, y1 + y2 ) = φ(x, y1 ) + φ(x, y2 ),

φ(αx, y) = αφ(x, y) = φ(x, αy). Equivalently, φ(α1 x1 + α2 x2 , β1 y1 + β2 y2 ) = φ(α1 x1 , β1 y1 + β2 y2 ) + φ(α2 x2 , β1 y1 + β2 y2 ) = φ(α1 x1 , β1 y1 ) + φ(α1 x1 , β2 y2 ) + φ(α2 x2 , β1 y1 ) + φ(α2 x2 , β2 y2 ) = α1 β1 φ(x1 , y1 ) + α1 β2 φ(x1 , y2 ) + α2 β1 φ(x2 , y1 ) + α2 β2 φ(x2 , y2 ). For the particular case of Z = F we will refer to a scalar-valued bilinear map φ : X ×Y → F either as a bilinear functional or as a bilinear form.

2.1 The Linear Space of Bilinear Maps As usual, let Z S denote the collection of all functions on a set S with values in a linear space Z, which is itself a linear space over the same field F . Let   b[X ×Y, Z] = φ ∈ Z X ×Y : φ is bilinear © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Kubrusly, Bilinear Maps and Tensor Products in Operator Theory, Universitext, https://doi.org/10.1007/978-3-031-34093-2 2

19

20

2. Bilinear Maps: Algebraic Aspects

denote the collection of all Z-valued bilinear maps on X ×Y. Bilinearity of elements φ : X ×Y → Z in b[X ×Y, Z] implies that b[X ×Y, Z] is a linear manifold of the linear space Z X ×Y of all Z-valued functions on the set S = X ×Y. Indeed, take arbitrary φ and φ from b[X ×Y, Z]. Since φ and φ are bilinear, it follows that for every x1 , x2 ∈ X and y ∈ Y, (φ + φ )(x1 + x2 , y) = φ(x1 + x2 , y) + φ (x1 + x2 , y) = φ(x1 , y) + φ(x2 , y) + φ (x1 , y) + φ (x2 , y) = (φ + φ )(x1 , y) + (φ + φ )(x2 , y). Similarly, for every x ∈ X , every y, y1 , y2 ∈ Y, and every α ∈ F , (φ + φ )(x, y1 + y2 ) = (φ + φ )(x, y1 ) + (φ + φ )(x, y2 ), (φ + φ )(αx, y) = α(φ + φ )(x, y) = (φ + φ )(x, αy). Thus φ + φ is a bilinear map, which means φ + φ lies in b[X ×Y, Z]. Also, since φ is bilinear, then for every x1 , x2 ∈ X , y ∈ Y. and β ∈ F , (β φ)(x1 + x2 , y) = φ(x1 + x2 , β y) = φ(x1 , β y) + φ(x2 , β y) = (β φ)(x1 , y) + (β φ)(x2 , y). Similarly, for every x ∈ X , every y, y1 , y2 ∈ Y, and every α, β ∈ F , (β φ)(x, y1 + y2 ) = (β φ)(x, y1 ) + (β φ)(x, y2 ), (β φ)(αx, y) = α(β φ)(x, y) = (β φ)(x, αy). So β φ is again a bilinear map, which means β φ lies in b[X ×Y, Z]. Hence b[X ×Y, Z] is a linear manifold of the linear space Z X ×Y. Therefore b[X ×Y, Z] is a linear space over F . Let R(φ) = range(φ) = φ(X ×Y) denote the range of φ ∈ b[X ×Y, Z], a subset of the linear space Z which in general is not a linear manifold of Z. Remark 2.1. On Restrictions of Bilinear Maps. Consider the restriction φ|M ×N : M ×N → Z of a bilinear map φ : X ×Y → Z to a Cartesian product of subsets M and N of the linear spaces X and Y. Two cases deserve attention. (a) Restriction to the Cartesian product of linear manifolds M and N of X and Y. A straightforward consequence of the definitions of linear manifold and bilinear map leads to the following result. The restriction of a bilinear map to a Cartesian product of linear manifolds is bilinear . That is, if M and N are linear manifolds of X and Y, then φ ∈ b[X ×Y, Z] implies φ|M×N ∈ b[M×N , Z]. (b) Restriction to the Cartesian product of a singleton with either X or Y. This goes back to the definition of bilinear maps, leading to linear sections φy = φ|X ×{y} in L[X , Z] and φx = φ|{x}×Y in L[Y, Z]. If the singleton is the

2.1 The Linear Space of Bilinear Maps

21

origin of one of the linear spaces, then it becomes the null linear manifold. Let 0 stand for the origin of X and also for the origin of Y. If φ is bilinear, then φ0 = φ|X ×{0} = O is the null transformation in L[X , Z] = b[X ×{0}, Z] and φ0 = φ|{0}×Y = O is the null transformation in L[Y, Z] = b[{0}×Y, Z]. Remark 2.2. On Composition of Bilinear Maps. As composition of a pair of linear transformations is a linear transformation, composition of a bilinear map with a linear transformation from both sides is a bilinear map. (a) On the one hand, composition of a bilinear map with a linear transformation is a bilinear map. That is, if X , Y, Z, Z  are linear spaces, then φ ∈ b[X ×Y, Z]

and

L ∈ L[Z, Z  ]

=⇒

L ◦ φ ∈ b[X ×Y, Z  ].

Indeed, since φ is bilinear and L is linear, then     (L ◦ φ) (x1 + x2 , y) = L φ(x1 + x2 , y) = L φ(x1 , y) + φ(x2 , y)     = L φ(x1 , y) + L φ(x2 , y) = (L ◦ φ) (x1 , y) + (L ◦ φ) (x2 , y) for every x1 , x2 ∈ X and y ∈ Y. Similarly, for every x ∈ X , y1 , y2 ∈ Y and α ∈ F , (L ◦ φ) (x, y1 + y2 ) = (L ◦ φ) (x, y1 ) + (L ◦ φ) (x, y2 ), (L ◦ φ) (α x, y) = α(L ◦ φ) (x, y) = (L ◦ φ) (x, α y). (b) On the other hand, composition of a pair of linear transformations with a bilinear map is a bilinear map. That is, for linear spaces X , X , Y, Y , Z, φ ∈ b[X ×Y, Z], L ∈ L[X , X ], K ∈ L[Y , Y] =⇒ φ(L(·), K(·)) ∈ b[X  ×Y , Z]. In fact, as is readily verified, for every u, x ∈ X  , v, y ∈ Y  and α, β, γ, δ ∈ F , φ(L(αu + β x), K(γ v + δy) = αγ φ(Lu, Kv) + αδ φ(Lu, Ky) + β γ φ(Lx, Kv) + β δ φ(Lx, Ky). Let X , Y, Z be linear spaces over F and take the Cartesian product X ×Y, which is not itself a linear space. Consider the linear space direct sum X ⊕ Y = (X ×Y, ⊕,  , F ) whose underlying set is X ×Y. Take ψ : X ⊕ Y → Z. Observe that ψ acts on the underlying set X ×Y of the linear space X ⊕ Y. Suppose ψ : X ×Y → Z is a bilinear map (i.e., ψ ∈ b[X ×Y, Z]). Set     b[X ⊕ Y, Z] = ψ ∈ Z X ⊕Y : ψ is bilinear = ψ ∈ Z X ⊕Y : ψ ∈ b[X ×Y, Z] , the linear space of all Z-valued bilinear maps on the linear space X ⊕ Y. In other words, since bilinearity only requires the algebraic structure of each linear space X and Y individually, and since the map ψ actually acts on the underlying set X ×Y of X ⊕ Y independently of the algebra equipping X ⊕ Y, its bilinearity does not depend on the algebra of the linear space X ⊕ Y. Thus b[X ⊕ Y, Z] = b[X ×Y, Z].

22

2. Bilinear Maps: Algebraic Aspects

Such an identity deserves a few words. The domain of a map in b[X ×Y, Z] has no algebraic structure, while the domain of a map in b[X ⊕ Y, Z] has. Proposition 2.C in Section 2.4 will discuss this point. So if some algebraic property, say, translation, is required in the domain of bilinear maps, then b[X ⊕ Y, Z] is the proper setup. This is illustrated in the proof of Proposition 6.F in Section 6.4. Remark 2.3. Nonlinearity and Quadratic Forms. Bilinear forms are closer to quadratic forms than to linear forms. (a) Even if a bilinear map ψ is regarded as an element of b[X ⊕ Y, Z], it is not a linear transformation on the linear space X ⊕ Y. In other words, ψ ∈ b[X ⊕ Y, Z]\L[X ⊕ Y, Z], where L[X ⊕ Y, Z] is the linear space of all linear transformations from the direct sum X ⊕ Y to the linear space Z. For instance, take any (x, y) ∈ X ⊕ Y for which ψ(x, y) = 0 in Z (i.e., (x, y) ∈ (X ⊕ Y)\N (ψ)) and compute ψ((x, y) ⊕ (x, y)) = ψ(2x, 2y) = 4ψ(x, y) = 2ψ(x, y) = ψ(x, y) + ψ(x, y). So even when acting on the linear space X ⊕ Y bilinear maps are not additive. (b) Moreover, the product of linear forms acting on X and Y is a bilinear form acting on X ×Y. In other words, if f ∈ X  = L[X , F ] and g ∈ Y  = L[Y, F ] are linear forms, then φ : X ×Y → F given by φ(x, y) = f (x) g(y) ∈ F

for every

(x, y) ∈ X ×Y

is clearly a bilinear form (i.e., φ ∈ b[X ×Y, F ]). In particular, if Y = X and f = g, then φ(x, y) = φ(y, x) and φ(x, x) = f (x)2 for every x, y ∈ X . There are, however, bilinear forms which are not the product of linear forms. Indeed, if a bilinear form on X ×X induces a positive quadratic form on a space X of dimension greater than one, then it is not the product of linear forms. For example, set X = Y, let F = R , and take an inner product · ; · : X ×X → R in the real linear space X . This is a (symmetric) bilinear form on X ×X which induces a positive quadratic form when evaluated at (x, x), namely, the norm squared, x ; x = x 2

for every

x ∈ X.

(Quadratic forms and inner product are discussed in Section 2.4). Suppose there exist f, g ∈ X  such that x 2 = x ; x = f (x) g(x) for every x in X . Since x = 0 if and only if f (x) g(x) = 0, then N (f ) ∪ N (g) = {0}. This implies N (f ) = N (g) = {0} or, equivalently, f and g are injective. But there is no injective linear functional on a linear space of dimension greater than 1 (i.e., a linear functional on a nonzero space is injective if and only if it acts on a onedimensional space — cf. Proposition 1.F in Section 1.4). So if dimX ≥ 2, then a real inner product (which is bilinear) is never the product of two linear forms. (c) There are, however, relations between bilinear maps and linear transformations. This is investigated in Theorem 2.6 below, and then again in Theorem 3.7, wherein tensor products will somehow rescue linearity from bilinearity.

2.2 Extension of Bilinear Maps

23

2.2 Extension of Bilinear Maps Bilinear maps as products of linear transformations extend bilinearly from Cartesian products of linear manifolds to Cartesian products of linear spaces, as linear transformations extend linearly from linear manifolds to linear spaces. Theorem 2.4. Let X and Y be linear spaces, let Z be an algebra, all over the same field, and let M and N be linear manifolds of X and Y respectively. If φ ∈ b[M×N , Z] is a bilinear map on M×N of the form φ(u, v) = Lu Kv for every (u, v) ∈ M×N with L ∈ L[M, Z] and K ∈ L[N , Z], then there exists a  y) = Lx  Ky  bilinear extension φ ∈ b[X ×Y, Z] of φ on X ×Y of the form φ(x,   for every (x, y) ∈ X ×Y with L ∈ L[X , Z] and K ∈ L[Y, Z]. Proof. Suppose Z is an algebra. Then take a bilinear map φ ∈ b[M×N , Z] such that φ(u, v) = Lu Kv for every (u, v) ∈ M×N . Since L ∈ L[M, Z] and  ∈ L[X , Z] K ∈ L[N , Z], they have linear extensions over X and Y, say, L    and K ∈ L[Y, Z] such that L|M = L and K|N = K (cf. Theorem 1.1). Thus  y) = Lx  Ky  for every (x, y) ∈ X ×Y, which take φ : X ×Y → Z given by φ(x,   M×N : M×N → Z is such that is bilinear (i.e., φ ∈ b[X ×Y, Z]). Moreover, φ|     φ|M×N (u, v) = L|M u K|N v = Lu Kv = φ(u, v) for every (u, v) ∈ M×N . The above extension is restricted to bilinear maps that are the product of two linear transformations with values in an algebra. In particular, it applies to bilinear forms that are the product of two linear forms (i.e., two linear functionals). The proof is an application of Theorem 1.1 on extensions of linear transformations. The next result, whose proof follows the argument in the proof of Theorem 1.1, generalizes Theorem 2.4. Theorem 2.5. Let X , Y and Z be linear spaces over the same field F and let M and N be nonzero linear manifolds of X and Y. If φ : M×N → Z is a bilinear map, then there exists a bilinear extension φ : X ×Y → Z of φ defined on the Cartesian product X ×Y of the whole spaces X and Y. Proof. Again, let Lat(X ) stand for the lattice of all linear manifolds of a linear space X . Take an arbitrary φ ∈ b[M×N , Z] and set    M×N , Φ = φ ∈ b[U ×V, Z] : U ×V ∈ Lat(X )×Lat(Y), M ⊆ U , N ⊆V, φ = φ| the collection of all bilinear maps φ from the Cartesian product of linear manifolds of X and Y to Z which are extensions of φ.  Note that Φ is nonempty (at least φ is there). Also, as a subcollection of C = M ×N ⊆ X ×Y Z M ×N (the collection of all maps into Z whose domain M ×N is included in X ×Y with M and N being subsets of X and Y), Φ is partially ordered in the extension  ordering ≤ and every chain {φγ } in Φ has a supremum γ φγ in C with       D(φγ ) and range R R(φγ ) (see, e.g., [52, domain D φγ = φγ = γ

γ

γ

γ

24

2. Bilinear Maps: Algebraic Aspects

Problem 1.17]). Since D(φγ ) ∈ Lat(X )×Lat(Y) (each φγ is a bilinear map on the Cartesian product of linear manifolds of X and Y), and since Lat(X ) and Lat(Y) are complete lattices (i.e., every subset of each of them has a supremum    and an infimum), it follows that D γ φγ = γ D(φγ ) lies in Lat(X )×Lat(Y).    Note: R γ φγ = γ R(φγ ) is a subset of Z (not in Lat(Z)).  Claim. The supremum γ φγ lies in Φ.   Proof of Claim. Take any (u1 , v1 ) and (u2 , v2 ) in D γ φγ ∈ Lat(X )×Lat(Y). Then (u1 , v1 ) ∈ D(φ1 ) for some φ1 ∈ {φγ } and (u2 , v2 ) ∈ D(φ2 ) for some φ2 ∈ {φγ }. Since {φγ } is a chain we get φ1 ≤ φ2 (or vice versa), which implies      D(φ1 ) ⊆ D(φ2 ). Thus φγ |  = φ2 . So φγ : D φγ → Z is bilinear. γ

D(φ2 )

γ

γ

(Indeed, since D(φ1 ) ⊆ D(φ2 ), (α1 u1 + α2 u2 , β1 v1 + β2 v2 ) ∈ D(φ2 ). Since each φγ is a bilinear map, φ2 (α1 u1 + α2 u2 , β1 v1 + β2 v2 ) = α1 β1 φ2 (u1 , v1 ) + α1 β2 φ2 (u1 , v2 ) + α2 β1 φ2 (u2 , v1 ) + α2 β2 φ2 (u2 , v2 ) for every α1 , α2 , β1 , β2 ∈ F .      However, since γ φγ |D(φ2 )= φ2 , we get γ φγ (α1 u1 + α2 u2 , β1 v1 + β2 v2 ) =         α1 β1 φγ (u1 , v1 )+α1 β2 φγ (u1 , v2 )+α2 β1 φγ (u2 , v1 )+α2 β2 φγ γ

γ

γ

γ

(u2 , v2 ) because φ2 is bilinear. Since (u1 , v1 ) and (u2 , v2 ) were arbitrarily taken      in D γ φγ ∈ Lat(X )×Lat(Y), thus γ φγ : D γ φγ → Z is bilinear.)

Moreover, since each φγ is such that φγ |M×N = φ, and since {φγ } is a chain,    it also follows that γ φγ |M×N = φ. Conclusion: γ φγ ∈ Φ. 

Therefore every chain in Φ has a supremum (and so an upper bound) in Φ. Thus according to Zorn’s Lemma, Φ contains a maximal element, say φ0 : U 0 ×V 0 → Y, where U 0 and V 0 are linear manifolds of X and Y such that M ⊆ U 0 ⊆ X and N ⊆ V 0 ⊆ Y. We will show that U 0 ×V 0 = X ×Y, and hence φ0 is a bilinear extension of φ over X ×Y. The proof goes by contradiction. Suppose U 0 ×V 0 = X ×Y so that U 0 = X , or V 0 = Y, or both. (a) To begin with, suppose U 0 = X . Take x1 ∈ X \U 0 so that x1 = 0. Consider the sum of U 0 and the one-dimensional linear manifold of X spanned by {x1 }, U 1 = U 0 + span {x1 },

(∗)

which properly includes M (as M ⊆ U 0 ⊂ U 1 ). Since U 0 ∩ span {x1 } = {0}, every x in U 1 has a unique representation as a sum of vectors in U 0 and in span {x1 }: for each x ∈ U 1 there is a unique pair (x0 , α) in U 0 × F such that x = x0 + αx1 . (Indeed, if x = x0 + αx1 = x0 + α x1 , then x0 − x0 = (α − α)x1 lies in N0 ∩ span {x1 } = {0} so that x0 = x0 and α = α — recall: x1 = 0.) Now let V 1 be a linear manifold of Y such that N ⊆ V 0 ⊆ V 1 ⊆ Y.

2.2 Extension of Bilinear Maps

25

(a1 ) First suppose V 1 = V 0 . Consider the map φ1 : U 1 ×V 1 → Z defined by φ1 (x, y) = φ0 (x0 , y)

for every

(x, y) ∈ U 1 ×V 1 = U 1 ×V 0 .

Since φ0 : U 0 ×V 0 → Z is bilinear and since U 0 ⊂ U 1 (proper inclusion), then φ1 is a bilinear map and φ0 = φ1 |U 0 ×V 0 , and therefore φ0 ≤ φ1 . Also, since M ⊆ U 0 ⊂ U 1 ⊆ X , N ⊆ V 0 = V 1 ⊆ Y, and φ = φ0 |M×N , then φ = φ0 |M×N = φ1 |M×N . So φ1 ∈ Φ, which contradicts the fact that φ0 is maximal in Φ (as φ0 = φ1 ). (a2 ) Next suppose V 0 = V 1 (i.e., V 0 ⊂ V 1 ). Take y1 ∈ V 1 \V 0 (again y1 = 0). Then in addition to (∗) consider the linear manifold V 1  defined by V 1  = V 0 + span {y1 },

(∗∗)

which satisfies N ⊆ V 0 ⊂ V 1  ⊆ V 1 ⊆ Y. In (∗) we saw that every x ∈ U 1 has a unique representation as x = x0 + αx1 for some x0 ∈ U 0 and α ∈ F . Similarly, from (∗∗) we see that every y ∈ V 1  has a unique representation as y = y0 + βy1 for some y0 ∈ V 0 and β ∈ F . Consider the map φ1 : U 1 ×V 1  → Z defined by φ1 (x, y) = φ0 (x0 , y0 ) + αβz

for every

(x, y) ∈ U 1 ×V 1 

for an arbitrary z ∈ Z. Again, since U 0 ⊂ U 1 and V 0 ⊂ V 1 (proper inclusions), φ1 is bilinear and φ0 = φ1 |U 0 ×V 0 , and hence φ0 ≤ φ1 , as φ1 inherits the bilinearity of φ0 . (Indeed, (i) φ1 (x + x , y) = φ0 (x0 + x0 , y) + (α + α )βz = (φ0 (x0 , y) + αβz) + (φ0 (x0 , y) + α βz) = φ1 (x , y) + φ1 (x , y), (ii) φ1 (δx, y) = φ0 (δx0 , y0 ) + δαβz = δ(φ0 (x0 , y0 ) + αβz) = δ φ1 (x, y)), (iii) φ1 (x, y + y  ) = φ0 (x0 , y0 + y0 ) + α(β + β  )z = (φ0 (x0 , y0 ) + αβz) + (φ0 (x0 , y0 ) + αβ  z) = φ1 (x, y) + φ1 (x, y  ), (vi) φ1 (x, δy) = φ0 (x0 , δy0 ) + αδβz = δ(φ0 (x0 , y0 ) + αβz) = δ φ1 (x, y), and hence φ1 : U 1 ×V 1  → Z is bilinear.) Since φ = φ0 |M×N is a bilinear map, and since M ⊆ U 0 ⊂ U 1 ⊆ X and N ⊆ V 0 ⊂ V 1  ⊆ V 1 ⊆ Y,

26

2. Bilinear Maps: Algebraic Aspects

φ = φ0 |M×N = φ1 |M×N .  φ1 ). So φ1 ∈ Φ, contradicting the fact that φ0 is maximal in Φ (as φ0 = (b) On the other hand, suppose V 0 = Y. Let U 1 be any linear manifold of X such that M ⊆ U 0 ⊆ U 1 ⊆ X . If U 0 = U 1 , then we get a case symmetrical to (a1 ). If U 0 ⊂ U 1 , then we get a case symmetrical to (a2 ). Both cases then lead to the same consequence as before. (c) Thus by (a) and (b) U 0 ×V 0 = X ×Y leads to a contradiction.



2.3 Identification with Linear Transformations Bilinear maps can also be extended by factoring them by the natural bilinear map through a tensor product space. This will be shown in Remark 3.8(a) (as a consequence of Theorem 3.7 — next chapter). Remark 3.8(a) will ensure bilinear extension out of linear extension, yielding a shorter proof for Theorem 2.5. It is, however, advisable to have a proof for bilinear extension independently of any further result, as was shown in Theorem 2.5. Yet there is still another way to prove Theorem 2.5. This will be shown in Remark 2.7(a) below, which depends on the following pivotal theorem. Theorem 2.6. Take an arbitrary triple (X , Y, Z) of linear spaces over the same field F . The linear spaces b[X ×Y, Z] and L[X , L[Y, Z]] are isomorphic: b[X ×Y, Z] ∼ = L[X , L[Y, Z]]. Proof. Take an arbitrary φ ∈ b[X ×Y, Z]. For each x ∈ X take the x-section φx ∈ L[Y, Z], which determines a map Lφ : X → L[Y, Z] given by Lφ x = φx

for every

x ∈ X,

for each

φ ∈ b[X ×Y, Z].

which is clearly linear: Lφ ∈ L[X , L[Y, Z]]

(Indeed, Lφ (x1 + x2 ) = φx1+x2 = φ(x1 + x2 , ·) = φ(x1 , ·)+φ(x2 , ·) = φx1+ φx2 = Lφ (x1 + x2 ) and Lφ (αx) = φαx = αφx = αLφ x for x, x1 , x2 ∈ X and α ∈ F .) Now consider the map  : b[X ×Y, Z] → L[X , L[Y, Z]] defined by (φ) = Lφ

for every

φ ∈ b[X ×Y, Z].

It is easy to verify that this is again a linear transformation:

 ∈ L b[X ×Y, Z], L[X , L[Y, Z]] . (In fact, Lφ+φ x = (φ + φ )x = (φ + φ )(x, ·) = φ(x, ·) + φ (x, ·) = φx + φx = Lφ x + Lφ x = (Lφ + Lφ ) x and Lαφ x = (αφ)x = αφx = αLφ x for every

2.3 Identification with Linear Transformations

27

x ∈ X and α ∈ F , and so (φ + φ ) = Lφ+φ = Lφ + Lφ = (φ) + (φ ) and (αφ) = Lαφ = αLφ = α(φ), for every φ, φ ∈ b[X ×Y, Z] and α ∈ F .) It is also readily verified that  is injective (for (φ) = 0 implies Lφ (x) = 0 and so φx = 0, for every x ∈ X , which leads to φ = 0, thus ensuring N () = {0}). To show that  is surjective, take an arbitrary L ∈ L[X , L[Y, Z]] and define φ : X ×Y → Z by φ(x, y) = (Lx) y ∈ Z, which is bilinear (as φy (·) = (L(·)) y lies in L[X , Z] for each y in Y and φx (·) = (Lx)(·) lies in L[Y, Z] for each x in X ), and hence Lφ (x) = φx = (Lx)(·) so that (φ) = Lφ = Lφ (·) = L(·)(·) = L, and therefore L ∈ R(), which implies R() = L[X , L[Y, Z]]. Thus  : b[X ×Y, Z] → L[X , L[Y, Z]] is an isomorphism.



Remark 2.7. Two Important Consequences of Theorem 2.6. (a) A second proof of Theorem 2.5 (based on Theorems 1.1 and 2.6). Take an arbitrary φ ∈ b[M×N , Z]. By Theorem 2.6 there exists a linear-space isomorphism  of b[M×N , Z] onto L[M, L[N , Z]] whose inverse takes each L = (φ) ∈ L[M, L[N , Z]] to φ = −1 (L) such that φ(u, v) = (Lu)v for every (u, v) in M×N where (Lu) ∈ L[N , Z] for each u ∈ M. (i) Apply Theorem 1.1 to extend L over X and get L∧ ∈ L[X , L[N , Z]] such that (L∧x) ∈ L[N , Z] for each x ∈ X and (L∧u) = L∧|M u = (Lu) ∈ L[N , Z] for every u ∈ M ⊆ X , and so φ(u, v) = (L∧u)v for every (u, v) ∈ M×N . (ii) Next for each x ∈ X apply Theorem 1.1 again to extend (L∧x) ∈ L[N , Z]   = (L ∧x) ∈ L[Y, Z]. This defines a linear transformation over Y and get Lx  L ∈ L[X , L[Y, Z]]. Indeed, the map L∧ : X → L[N , Z] is clearly a linear transformation, and the extension operation is also linear (regarded as a map  of L[N , Z] into the quotient space L[Y, Z] /M consisting of the equivalence  of all possible linear extensions L  of each linear transformation L in classes [L] ∧ L[Y, Z] as in Remark 1.2). Then L (x1 + x2 ) = (L∧x1 ) + (L∧x2 ) ∈ L[N , Z],     1 + x2 ) = (L∧(x and so L(x 1 + x2 )) = (L∧x1 ) + (L∧x2 ) = (L∧x1 ) + (L∧x2 ) =     Lx1 + Lx2 for x1 , x2 ∈ X . Similarly, L(αx) = α Lx for x ∈ X and α ∈ F .  ∈ L[X , L[Y, Z]] we get (Lx)(y)  (iii) So for L ∈ Z for every (x, y) ∈ X ×Y and    also (Lu)(v) = (L∧u)(v) = (L∧u)|N (v) = (L∧u)(v) = (Lu)(v) ∈ Z for every  (u, v) ∈ M×N ⊆ X ×Y. Thus φ(u, v) = (Lu)(v) for every (u, v) ∈ M×N .  y) = (Lx)(y)  (iv) Finally, set φ(x, for every (x, y) in X ×Y. This defines a map  φ : X ×Y → Z which is clearly bilinear (the same argument as in the proof of  : X → L[N , Z] is linear). Therefore φ ∈ b[X ×Y, Z] is an Theorem 2.6 once L  M×N = φ. extension of φ ∈ b[M×N , Z] over X ×Y since φ| 

(b) Another consequence (straightforward but helpful) of Theorem 2.6 is this. Take the permutation map Π : Y×X → X ×Y such that Π(y, x) = (x, y) and consider a transformation ℘ : b[X ×Y, Z] → b[Y×X , Z] given by ℘ (φ) = φ ◦ Π.

28

2. Bilinear Maps: Algebraic Aspects

In other words, ψ = ℘ φ in b[Y×X , Z] is such that ψ(y, x) = φ(Π(y, x)) = φ(x, y) for every φ in b[X ×Y, Z]. As is readily verified, ℘ is an isomorphism between the linear spaces b[X ×Y, Z] and b[Y×X , Z]. Thus by Theorem 2.6 L[Y, L[X , Z]] ∼ = b[Y×X , Z] ∼ = b[X ×Y, Z] ∼ = L[X , L[Y, Z]]. Now let X  = L[X , F ] and Y  = L[Y, F ] be the algebraic duals of X and Y. Set Z = F above and get an isomorphic representation of the bilinear forms: L[Y, X  ] ∼ = b[Y×X , F ] ∼ = b[X ×Y, F ] ∼ = L[X , Y  ].

2.4 Additional Propositions Let X , Y, Z be nonzero linear spaces over the same field F , let ∼ = stand for isomorphism, and let b[X ×Y, Z] and L[X , Z] denote the linear spaces of all bilinear maps from X ×Y to Z and of all linear transformations of X into Z. Proposition 2.A. b[X ×Y, F ] = {0} if and only if either X = {0} or Y = {0}. Under what circumstances is a bilinear map injective? [15, Exercise 1.10]. For an arbitrary φ ∈ b[X ×Y, Z] consider the following subset of X ×Y : N (φ) = {(x, y) ∈ X ×Y : φ(x, y) = 0}. Let N (φy ) and N (φx ) be the kernels of the sections φy in L[X , Z] and φx in L[Y, Z], which are linear manifolds of the linear spaces X and Y, respectively. A bilinear map φ : X ×Y → Z is injective (i.e., one-to-one) if for every z in the range R(φ) ⊆ Z of φ there is a unique pair (x, y) in X ×Y such that z = φ(x, y). Proposition 2.B. Take an arbitrary φ ∈ b[X ×Y, Z].       (a) N (φ) = x∈X {x}×N (φx ) ∪ y∈Y N (φy )×{y} . (b) If φ is injective, then N (φ) = {(0, 0)}. (c) If one of the linear spaces X or Y is nonzero, then N (φ) = {(0, 0)}. Proposition 2.C. Take an arbitrary bilinear map φ ∈ b[X ×Y, Z]. Consider the linear space direct sum X ⊕ Y = (X ×Y, ⊕,  , F ). Let ΠX : X ⊕ Y → X ⊕ Y and ΠY : X ⊕ Y → X ⊕ Y be given by ΠX (x, y) = (0, y) and ΠY (x, y) = (x, 0) for every (x, y) ∈ X ⊕ Y. These are linear projections with R(ΠX ) = {0} ⊕ Y and R(ΠY ) = X ⊕ {0}. Since X ⊕ {0} and {0} ⊕ Y are naturally isomorphic to X and Y, write (i.e., identify) ΠX (x, y) = y and ΠY (x, y) = x for every (x, y) ∈ X ⊕ Y. The map ψ : X ⊕ Y → Z given by ψ(x, y) = φ(ΠY (x, y), ΠX (x, y)) = φ(x, y) ∈ Z, for every x ∈ X and y ∈ Y, is bilinear (i.e., ψ ∈ b[X ⊕ Y, Z]). Moreover,

2.4 Additional Propositions

29

ψ((x1 , y1 ) ⊕ (x2 , y2 )) + ψ((x1 , y1 )  (x2 , y2 )) = 2ψ(x1 , y1 ) + 2ψ(x2 , y2 ), ψ(α(x, y)) = α2 ψ(x, y), for every x, x1 , x2 ∈ X , every y, y1 , y2 ∈ Y, and every α ∈ F , where the symbol  means (x1 , y1 )  (x2 , y2 ) = (x1 , y1 ) ⊕ (−x2 , −y2 ) ∈ X ⊕ Y. In particular , ψ((x, y) ⊕ (x, y)) = ψ(2(x, y)) = 4ψ(x, y). Proposition 2.D. Take φ ∈ b[X ×Y, Z], let M and N be linear manifolds of X and Y, consider the quotient spaces X /M and Y/N , and take the quotient maps πX ∈ L[X , X /M] and πY ∈ L[Y, Y/N ]. Now consider both sections φv = φ(·, v) in L[X , Z] for each v ∈ N and φu = φ(u, ·) in L[Y, Z] for each u ∈ M. Next take the linear manifold R = span R(φ) ⊆ Z and the quotient map πZ in L[Z, Z/R]. Since πZ is surjective, define ψ : X /M×Y/N → Z/R by ψ(πX x, πY y) = πZ φ(x, y) for every ([x], [y]) = (πX x, πY y) ∈ X /M×Y/N . The map ψ is bilinear . Take φ ∈ b[X ×Y, Z]. Let N (φy ) and N (φx ) be the kernels of the linear transformations φy ∈ L[X , Z] and φx ∈ L[Y, Z]. Consider the linear manifolds N (φy ) ⊆ X and NX (φ) = N (φx ) ⊆ Y. N Y (φ) = y∈Y

x∈X

Proposition 2.E. Take φ ∈ b[X ×Y, Z]. If M = N Y (φ), N = NX (φ), and if ψ : X /M×Y/N → Z/R is the bilinear map defined in Proposition 2.D, then N Y (ψ) = {0}

and

NX (ψ) = {0}.

The next propositions are isolated restatements of part of the isometric chain in Remark 2.7(b) as they will be required in the sequel. Proposition 2.F.

b[X ×Y, Z] ∼ = b[Y×X , Z].

Proposition 2.G.

L[X , L[Y, Z]] ∼ = L[Y, L[X , Z]].

We have already seen some examples of bilinear maps and bilinear forms. For instance, the product of linear forms (i..e., φ(x, y) = f (x) g(y)) and the inner product in any real inner product space (i.e., (x, y) → x ; y ∈ R ) are common examples of bilinear forms, as we saw in Remark 2.3. The real inner product offers a large source of concrete examples of bilinear forms. The product of a linear transformation with a linear form is a bilinear map (e.g., (x, y) → f (x) Ky is a bilinear map whenever f is a linear form and K is a linear transformation). If Z is an algebra, then the product of linear transformations is a bilinear map (i.e., φ(x, y) = Kx Ky is a bilinear map if L, K are linear transformations taking values on the same algebra Z — as in Theorem 2.4). Also, the composition of a bilinear map with a linear transformation

30

2. Bilinear Maps: Algebraic Aspects

  is a bilinear map (i.e., (x, y) → L φ(x, y) is bilinear if φ is a Z-valued bilinear map and L is a linear transformation on Z), and the composition of two  linear transformations with a bilinear map is a bilinear map (i.e., (x, y) → φ Lx, Ky is bilinear if φ is bilinear and L, K are linear transformations), as we saw in Remark 2.2. Moreover, since b[X ×Y, Z] is a linear space, every

n(finite) linear combination of bilinear maps yields a bilinear map (x, y) → i=1 αi φi (x, y). The following propositions exhibit four more classical examples. Proposition 2.H. The evaluation map φ : L[X , Y]×X → Y defined by φ(L, x) = Lx for every L ∈ L[X , Y] and every x ∈ X is bilinear . Proposition 2.I. The convolution map (f, g) → f  g is bilinear. In other words, if L1 denotes the linear space of all integrable 1 1 1 real-valued functions on the  real line, then the map φ : L ×L1 → L given by φ(f, g)(τ ) = (f  g)(τ ) = f (τ − t) g(t) dt for every f, g ∈ L is a bilinear . Proposition 2.J. The composition map (L, F ) → L ◦ F is bilinear. Precisely, let X , Y be linear spaces (over the same field, of course) and let S be a nonempty set. Take the linear space L[X , Y] of all linear transformations of X into Y, and the linear spaces X S and Y S of all functions from S to X and to Y, respectively. The map φ : L[X , Y]×X S → Y S such that φ(L, F ) = L ◦ F for every L ∈ L[X , Y] and F ∈ X S is bilinear . Proposition 2.K. Take the linear space F S of all F -valued functions on a nonempty set S. Let Y be a linear space (over the field F ) and take the linear space Y S of all Y-valued functions on S. The map φ : F S ×Y → Y S defined by φ(f, y) = f (·)y for every f ∈ F S and every y ∈ Y is bilinear. In particular, by setting S = X (a linear space over F ) and replacing F S by X  ⊂ F X, then the bilinear map φ : X  ×Y → Y X such that φ(f, y) = f (·)y for f ∈ X  and y ∈ Y is such that R(φ) ⊆ L[X , Y] (i.e., x ∈ X → f (x)y ∈ span {y} ⊆ Y is linear for every y ∈ Y and every f ∈ X  ). In other words, φ ∈ b[X  ×Y, L[X , Y]]. The next propositions exhibit four special examples required in Chapter 4. Proposition 2.L. For each pair (x, y) in the Cartesian product X ×Y, consider the transformation Lx,y : b[X ×Y, Z] → Z defined by Lx,y (ψ) = ψ(x, y) ∈ Z

for every

ψ ∈ b[X ×Y, Z].

2.4 Additional Propositions

31

(a) Each Lx,y is linear : Lx,y ∈ L[ b[X ×Y, Z], Z] for each (x, y) ∈ X ×Y. Now consider the map θ : X ×Y → L[ b[X ×Y, Z], Z] given by θ(x, y) = Lx,y ∈ L[ b[X ×Y, Z], Z]

(x, y) ∈ X ×Y.

for every

(b) This θ is bilinear : θ ∈ b[X ×Y, L[ b[X ×Y, Z], Z]]. The particular case of Z = F leads to θ ∈ b[X ×Y, b[X ×Y, F ] ]. Proposition 2.M. For each pair (x, y) in the Cartesian product X ×Y, consider the map ϑx,y : X  ×Y  → F defined by ϑx,y (f, g) = f (x) g(y) ∈ F

(f, g) ∈ X  ×Y  .

for every

(a) Each ϑx,y is bilinear : ϑx,y ∈ b[X  ×Y  , F ] for each (x, y) ∈ X ×Y. Now consider the map θ : X ×Y → b[X  ×Y  , F ] given by θ (x, y) = ϑx,y ∈ b[X  ×Y  , F ]

(x, y) ∈ X ×Y.

for every

(b) This θ is bilinear : θ ∈ b[X ×Y, b[X  ×Y  , F ]]. Proposition 2.N. For each pair (x, y) in the Cartesian product X ×Y, consider the map Lx,y : X  → Y defined by Lx,y (f ) = f (x)y

for every

f ∈ X .

(a) Each Lx,y is linear : Lx,y ∈ L[X , Y] for each f ∈ X  . Now consider the map θ : X ×Y → L[X  , Y] given by θ (x, y) = Lx,y ∈ L[X  , Y]

for every

(x, y) ∈ X ×Y.

(b) This θ is bilinear : θ ∈ b[X ×Y, L[X  , Y]]. Proposition 2.O. For each pair (x, y) in the Cartesian product X ×Y, con sider the map L x,y : Y → X defined by L x,y (g) = g(y)x

for every

g ∈ Y .

   (a) Each L x,y is linear : Lx,y ∈ L[Y , X ] for each g ∈ Y .

Now consider the map θ : X ×Y → L[Y  , X ] given by  θ (x, y) = L x,y ∈ L[Y , X ]

for every

(x, y) ∈ X ×Y.

(b) This θ is bilinear : θ ∈ b[X ×Y, L[Y  , X ]]. We know from Theorem 2.5 (or Remark 2.7(a)) that bilinear maps have a bilinear extension. This ensures that conditions (a) and (b) in the next proposition hold when the sets M and N are replaced by linear manifolds M and N .

32

2. Bilinear Maps: Algebraic Aspects

Proposition 2.P. Let M and N be nonempty subsets of X and Y, respectively. Take an arbitrary map φ : M ×N → Z of the Cartesian product of M and N into a linear space Z. There exists a bilinear map φ ∈ b[X ×Y, Z] such  M ×N = φ if and only if the following two conditions hold true: that φ|

(a) i αi ui = 0 for a finite

subset {ui } of M and a similarly indexed set of scalars {αi } implies i αi φ(ui , v) = 0 for every v ∈ N , and

(b) j βj vj = 0 for a finite

subset {vj } of N and a similarly indexed set of scalars {βj } implies j βj φ(u, vj ) = 0 for every u ∈ M . A bilinear form φ : X ×X → F from the Cartesian product of a linear space X with itself to a field F (in our case F is either R or C ) is symmetric if φ(x, y) = φ(y, x) for every x, y ∈ X . A quadratic form of a bilinear form is a functional φ : X → F obtained from a symmetric bilinear form φ as follows: φ (x) = φ(x, x)

for every

x ∈ X.

A symmetric bilinear form φ can be written in terms of the corresponding quadratic form φ as (see, e.g., [30, Exercise 6, Section 23], [6, Problem 3.P])   for every x, y ∈ X . φ(x, y) = 14 φ (x + y) − φ (x − y) If F = R, then the above identity is referred to as the real polarization identity. A sesquilinear form σ : X ×Y → F is a functional from the Cartesian product of linear spaces X and Y to a field F (R or C ) such that one section is linear and the other is conjugate linear — hence the terminology: “sesqui” means “one-and-a-half”. Thus we define a sesquilinear form as being additive in both arguments, homogeneous in the first argument, and conjugate homogeneous in the second argument. Therefore the y-section σ y = σ(·, y) : X → F is a linear form for every y ∈ Y, and the complex conjugate of the x-section σ x = σ(x, ·) : Y → F is a linear form for every x ∈ X . That is, σ y ∈ X  = L[X , F ]

and

σ x ∈ Y  = L[Y, F ].

A sesquilinear form σ : X ×X → F from the Cartesian product of a linear space X with itself to a field F is Hermitian symmetric if σ(x, y) = σ(y, x) for every x, y ∈ X . A quadratic form of a sesquilinear form, σ : X → F , is a functional obtained from a Hermitian symmetric sesquilinear form σ as follows: σ (x) = σ(x, x)

for every

x ∈ X.

If F = C , then the Hermitian symmetric sesquilinear form σ can be written in terms of the corresponding quadratic form σ according to the expression   σ(x, y) = 14 σ (x+y)−σ (x−y)+iσ (x+iy)−iσ (x−iy) for every x, y ∈ X. In this case the above identity is called the complex polarization identity. A quadratic form of a Hermitian symmetric sesquilinear form is real-valued (i.e., σ (x) = σ(x, x) ∈ R for every x ∈ X — see, e.g., [6, Problem 3.P]).

2.4 Additional Propositions

33

If F = R , then the notions of bilinear and sesquilinear forms, and of symmetric bilinear and Hermitian symmetric sesquilinear forms, coincide. An inner product · ; · : X ×X → F defined on the Cartesian product X×X of a linear space X with itself to a field F (either R or C ) is a Hermitian symmetric sesquilinear form that induces a positive quadratic form (i.e., σ (x) = σ(x, x) = x ; x > 0 for every 0 = x ∈ X ), and in this case the square root of the positive quadratic form generated by the inner product is a norm on X (i.e., σ (x) = σ(x, x) = x ; x = x 2 for every x ∈ X ). The above paragraphs on symmetric bilinear forms, quadratic forms, sesquilinear forms, Hermitian symmetric sesquilinear forms, positive quadratic forms, and inner product constitute a basis for the discussion in Remark 2.3(b). We will return to inner product in Section 5.4. The next proposition deals with restrictions of sesquilinear forms in light of the results for restrictions of bilinear maps in Proposition 2.P. Proposition 2.Q. Let S be a subset of X spanning X (i.e., span S = X ) and take an F -valued function σ : X ×X → F on the Cartesian product of X with itself. Take the subset S×S of X ×X . There exists a sesquilinear form |S×S = σ if and only if the following condition holds: σ  : X ×X → F such that σ

i αi si = 0 for

a finite subset {si } of S and a finite set of scalars {αi } implies i αi σ(si , s) = 0 and i αi σ(si , s) = 0 for every s ∈ S. In this case, σ  is the unique extension of σ. Also, σ  is Hermitian symmetric if and only if σ is (i.e., if and only if σ(r, s) = σ(s, r) for every r, s ∈ S). Notes: Bilinear maps play the central role in defining tensor products, the topic of the next chapter. The present section focuses on basic additional results and offers a collection of classical examples of bilinear maps acting on the Cartesian product of linear spaces. Proposition 2.A, whose proof follows easily from Proposition 1.G in Section 1.4, is fundamental. Proposition 2.B shows that there is no nontrivial injective bilinear map. Proof of Proposition 2.A. Suppose b[X ×Y, F ] = {0}. Set φf,g =f ·g ∈ b[X ×Y, F ] for each pair f ∈ X  and g ∈ Y  . Fix an arbitrary f0 ∈ X  . If Y = {0}, then for every 0 = y ∈ Y there exists gy ∈ Y  such that gy (y) = 0 by Proposition 1.G(ii). If φf0 ,gy (x, y) = f0 (x)gy (y) = 0 for every y ∈ Y, then f0 (x) = 0 for every x ∈ X so that f0 = 0. Since f0 is arbitrary, this implies that every f ∈ X  is null, and so X = {0} by Proposition 1.G(ii) again. The converse is trivial. 

Proof of Proposition 2.B. Recall: φy (x) = φ(x, y) = φx (y) for every (x, y) in X ×Y. Take an arbitrary x ∈ X . If y ∈ N (φx ), then φx (y) = 0. So φ(x, y) = 0 every x ∈ X . Therefore and hence x) ⊆ N  (x, y) ∈ N (φ). Thus {x}×N (φ  (φ) for  y {x}×N (φ N (φ ) ⊆ N (φ). Similarly, )×{y} ⊆ N (φ). Then x x∈X y∈Y  x∈X

   {x}×N (φx ) ∪

y∈Y

  N (φy )×{y} ⊆ N (φ).

34

2. Bilinear Maps: Algebraic Aspects

Conversely, if (x, y) ∈ N (φ), then for this x ∈ X we get y ∈ N  (φx ) so that  {x}×N (φx ) . is a subset of (x, y) lies in {x}×N (φx ), which trivially x∈X    Symmetrically, (x, y) also lies in y∈Y N (φy )×{y} . This proves (a). Item (b) is readily verified since φ(0, 0) = 0. Item (c) follows from (a) since N (φ) = {(0, 0)} implies X = Y = {0} by (a). In fact, if N (φ) = {(0, 0)}, then φ(x, y) = 0 for every y ∈ Y whenever x = 0. So φ(x, 0) = φx (0) = 0 for every 0 = x ∈ X , which is a contradiction since φx is a linear transformation for each x ∈ X . Thus N (φ) = {(0, 0)} implies X = {0}. Symmetrically, N (φ) = {(0, 0)} implies Y = {0}. This gives another proof for (c). 

Propositions 2.C and 2.D deal with composition of bilinear maps with respect to direct sum (extending the discussion in Remark 2.3) and quotient spaces, respectively (see, e.g., [27, Problem 1.2 and Section 1.2]). (The expression ψ(x, y) = φ(x, y) in Proposition 2.C may seem tautological, but it isn’t: φ((x1 , y1 ) + (x2 , y2 )) is meaningless as there is no addition in X ×Y, but ψ((x1 , y1 ) ⊕ (x2 , y2 )) is well-defined in X ⊕ Y — compare the last expression in Proposition 2.C with Remark 2.3(a)). For Proposition 2.E see, e.g., [27, Problem 1.5]. As said before, Propositions 2.F and 2.G are isolated restatements picked up from the chain of isomorphisms in Remark 2.7(b) as they will be required in the sequel, which are immediate consequences of using the permutation Π : Y×X → X ×Y in connection with Theorem 2.6. The examples in Propositions 2.H, 2.I and 2.K (see, e.g., [27, Problem 1.3.3] and [15, Sections 1.3(2), 1.3(4), 1.3(5), p. 9]), and in Proposition 2.J, are also readily verified. Propositions 2.L to 2.O will be verified in Section 4.3 and applied in Theorem 4.3 and Corollaries 4.5, 4.7, and 4.8. Proposition 2.P is a counterpart for the Cartesian product of sets of the extension of bilinear maps for the Cartesian product of linear manifolds in Theorem 2.5 (cf. [5, Problem 2.R]). Proposition 2.Q is a version of it concerning sesquilinear forms (cf. [5, Problem 2.R]).

Suggested Readings Brown and Pearcy [5] Defant and Floret [15] Greub [27]

Halmos [30] MacLane and Birkhoff [69] Roman [81]

3 Algebraic Tensor Product

This chapter presents a unified view of algebraic tensor products wherein the so-called universal property is taken as an axiomatic starting point, rather than as a theorem for a specific realization, and this approach leads to an abstract notion of algebraic tensor products of linear spaces. The concrete standard forms will be shown in the next chapter to be interpretations of the axiomatic formulation. Tensor products are axiomatically defined as follows.

3.1 Tensor Product of Linear Spaces Definition 3.1. Let X and Y be nonzero linear spaces over the same field F . A tensor product of X and Y is a pair (T, θ) consisting of a linear space T over F and a bilinear map θ : X ×Y → T fulfilling the following two axioms. (a) The range R(θ) = θ(X ×Y) of the bilinear map θ ∈ b[X ×Y, T ] spans T : span R(θ) = T

(i.e., span θ(X ×Y) = T ).

(b) If φ ∈ b[X ×Y, Z] is an arbitrary bilinear map into an arbitrary linear space Z over F , then there is a linear transformation Φ ∈ L[T, Z] for which φ = Φ ◦ θ. That is, the diagram φ

X ×Y −−−→ Z  ⏐  ⏐Φ θ  T commutes, and so the map θ factors every bilinear map through T . A word on terminology. The linear space T is referred to as a tensor product space of X and Y associated with the bilinear map θ, which in turn is referred © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Kubrusly, Bilinear Maps and Tensor Products in Operator Theory, Universitext, https://doi.org/10.1007/978-3-031-34093-2 3

35

36

3. Algebraic Tensor Product

to as the natural bilinear map (or simply the natural map) associated with the linear space T = span R(θ) = span θ(X ×Y). Consider the natural bilinear map θ : X ×Y → T = span θ(X ×Y) associated with T. The value θ(x, y) of θ at a pair (x, y) in X ×Y is denoted by x ⊗ y, x ⊗ y = θ(x, y) ∈ T

for every

(x, y) ∈ X ×Y,

and x ⊗ y is called a single tensor (or a simple tensor , or an elementary tensor , or a decomposable element, or a single tensor product of x and y) in the tensor product space T. Take (x, y) ∈ X ×Y and α ∈ F arbitrarily. Since θ is a bilinear map, αθ(x, y) = θ(αx, y) = θ(x, αy); in particular, θ(x, y) = θ(αx, α1 y) = θ( α1 x, αy) for α = 0. Thus a multiple of a single tensor is again a single tensor, α(x ⊗ y) = (αx) ⊗ y = x ⊗ (αy), and the representation of any single tensor is not unique. For instance, x ⊗ y = (αx) ⊗ ( α1 y) = ( α1 x) ⊗ (αy) for every nonzero α ∈ F . In fact, finite sums of single tensors have uncountably many representations. For example, x ⊗ y = αx ⊗ α1 y+u ⊗ v−βu ⊗ β1 v, where u ⊗ v−βu ⊗ β1 v = 0 ⊗ 0, for every x, u ∈ X , every y, v ∈ Y, and every nonzero α, β ∈ F . So the origin 0 ∈ R(θ) ⊆ span R(θ) = T of the linear space T can be represented by 0 ⊗ 0 = x ⊗ 0 = 0 ⊗ y for every x ∈ X and every y ∈ Y. Also (x1 + x2 ) ⊗ (y1 + y2 ) = (x1 ⊗ y1 ) + (x1 ⊗ y2 ) + (x2 ⊗ y1 ) + (x2 ⊗ y2 ) for every x1 , x2 ∈ X and y1 , y2 ∈ Y, since x ⊗ y is the value of a bilinear map θ : X ×Y → Z at a pair (x, y) ∈ X ×Y, where the above sums are linear-space additions, in X and Y on the left-hand side and in Z on the right-hand side. Remark 3.2. Existence and Interpretations. (a) Given a pair (X , Y) of linear spaces over the same field, there exists a linear space T and a bilinear map θ ∈ b[X ×Y, T ] such that the pair (T, θ) is a tensor product of X and Y as in Definition 3.1. In other words, for every pair (X , Y) of linear spaces over the same field there are examples (or interpretations) of tensor product spaces T of X and Y. (b) Some of these will be exhibited in Chapter 4. So the proof of the existence of tensor products will be postponed till then. (The simplest example is presented in Proposition 4.B.) If (T, θ) and (T , θ ) are interpretations of tensor products for a given pair (X , Y) of linear spaces, then these interpretations are isomorphic, as will be shown in Theorem 3.5 and Corollary 3.6. (c) For every bilinear map φ ∈ b[X ×Y, Z] there exists a linear transformation Φ ∈ L[T, Z] such that φ = Φ ◦ θ whenever (T, θ) is a tensor product of X and Y (Definition 3.1). If there is a bilinear map φ for which the associated linear transformation Φ is injective, then Φ : T → Z embeds the tensor product space T in the linear space Z. (This will be shown in Corollary 4.10; next chapter.)

3.1 Tensor Product of Linear Spaces

37

The following two theorems and their corollaries establish the first properties of tensor products of linear spaces, viz., representation and uniqueness. Theorem 3.3. An element of a tensor product space is represented as a finite sum of single tensors (and such a representation is not unique):  ∈T ⇐⇒ = xi ⊗ yi (a finite sum). i

n Proof. This follows from axiom (a) in Definition 3.1. Indeed, i=1 xi ⊗ yi = n n i=1 θ(xi , yi ) lies in T for every finite collection {(xi , yi )}i=1 of pairs (xi , yi ) in X ×Y since R(θ) ⊆ T and T is a linearspace. Conversely, if  lies in T = n n n span R(θ), then  = i=1 αi θ(xi , yi ) = i=1 θ(αi xi , yi ) = i=1 (αi xi ) ⊗ yi because θ is bilinear and θ(αi xi , yi ) is the single tensor (αi xi ) ⊗ yi . As we saw before, the representation of a finite sum of single tensors is not unique.  Corollary 3.4. If T is a tensor product space, then the linear transformation Φ ∈ L[T, Z] associated with each bilinear map φ ∈ b[X ×Y, Z] in axiom (b) of Definition 3.1 is unique. Proof. Consider Definition 3.1. If for a given φ in b[X ×Y, Z] there are Φ1 and Φ2 in L[T, Z] such  that Φ1 ◦ θ = Φ2 ◦ θ = φ, then (Φ1 − Φ2 ) ◦ θ = 0, so that (Φ1 − Φ2 )θ(x, y) = 0 for every x, y ∈ X ×Y. Consequently (Φ1 − Φ2 )() =   (Φ1 − Φ2 ) i θ(xi , yi ) = i (Φ1 − Φ2 ) θ(xi , yi ) = 0 for every  ∈ T (since Φ1 − Φ2 is linear and  = i θ(xi , yi ) by Theorem 3.3). Hence Φ1 = Φ2 .  The natural bilinear map θ associated with a tensor product space T is unique and, conversely, the tensor product space T associated with a natural bilinear map θ is unique. Thus θ is the unique bilinear map that factors every bilinear map through a tensor product space T . This is shown in the next theorem and its corollary. All linear spaces are over the same field. Theorem 3.5. If (T, θ) and (T , θ ) are tensor products of linear spaces X and Y, then there is a unique isomorphism Θ ∈ L[T , T ] such that Θ ◦ θ = θ. That is, Θ is such that the following diagram commutes: θ

X ×Y −−−→ T  ⏐  ⏐Θ θ  T . Proof. Let (T, θ) and (T , θ ) be tensor products of linear spaces X and Y. Take an arbitrary bilinear map φ : X ×Y → Z into any linear space Z. Thus according to Definition 3.1 there are linear transformations Φ : T → Z and Φ : T  → Z such that φ = Φ ◦ θ = Φ ◦ θ, which means the diagrams

38

3. Algebraic Tensor Product φ

φ

X ×Y −−−→ Z  ⏐  ⏐Φ θ 

X ×Y −−−→ Z  ⏐   ⏐Φ θ 

and

T

T

commute. Since θ : X ×Y → T  and θ : X ×Y → T are bilinear maps, Definition 3.1 again ensures the existence of linear transformations Θ : T → T  and Θ : T  → T such that θ = Θ ◦ θ and θ = Θ◦ θ, which means the diagrams θ

θ

X ×Y −−−→ T   ⏐   ⏐Θ θ 

X ×Y −−−→ T  ⏐  ⏐Θ θ 

and

T

T commute. Hence θ = (Θ ◦ Θ) ◦ θ

and

θ = (Θ ◦ Θ ) ◦ θ.

Let IS be the identity function on an arbitrary set S. By the above equations, Θ◦Θ|R(θ ) = IR(θ ) : R(θ ) → R(θ )

and

Θ◦Θ |R(θ) = IR(θ) : R(θ) → R(θ).

Since Θ and Θ are linear transformations (and so are their compositions), and since span R(θ) = T and span R(θ ) = T  by Definition 3.1, then we get Θ ◦ Θ = Θ ◦ Θ| spanR(θ ) = I spanR(θ ) = IT  , Θ ◦ Θ = Θ ◦ Θ | spanR(θ) = I spanR(θ) = IT . Hence each linear transformation Θ and Θ is the inverse of the other (thus  isomorphisms), and Θ (and so Θ ) is unique according Corollary 3.4. Corollary 3.6. A tensor product of linear spaces is unique up to an isomorphism in the following sense. If (T, θ) and (T , θ ) are tensor products of the same pair of linear spaces, then (T, θ) = (ΘT , Θ θ ) for an isomorphism Θ in L[T  , T ]. In particular, two tensor product spaces of the same pair of linear spaces coincide if and only if the natural bilinear maps coincide. Proof. This is a restatement of Theorem 3.5. Uniqueness up to isomorphism is immediate from Theorem 3.5. By uniqueness of the isomorphism Θ in Theorem  3.5, if T = T  then Θ is the identity map and so θ = θ, and vice versa. There exists a tensor product for every pair of linear spaces (Remark 3.2). A tensor product for a pair of linear spaces is unique up to an isomorphism (Corollary 3.6). Then for any pair (X , Y) of linear spaces it is common to write T = X ⊗ Y. So one refers to X ⊗ Y as the tensor product space (and to (X ⊗ Y, θ) as the tensor product) of any pair of linear spaces X and Y. Thus if M and N are linear manifolds of X and Y, then M ⊗ N is a tensor product space. Actually,

3.2 Further Properties of Tensor Product Spaces

39

M ⊗ N is a linear manifold of X ⊗ Y.  (Indeed, if  = i ui ⊗ vi ∈ M ⊗ N ⊆ X ⊗ Y for ui ∈ M ⊆ X and vi ∈ N ⊆ Y, then so does the sum of two such elements, as well as its product by a scalar.) Moreover, it will be shown in Theorem 3.13 that M ⊗ N = span R(θ|M×N ).

3.2 Further Properties of Tensor Product Spaces Each bilinear map on the Cartesian product X ×Y is in a one-to-one correspondence with a linear transformation on the tensor product space X ⊗ Y. In fact, the linear space of all bilinear maps b[X ×Y, Z] is isomorphic to the linear space of all linear transformations L[T, Z]. Thus a crucial property of a tensor product (T, θ) is to linearize bilinear maps as in Theorem 3.7 below (and this is done via factorization by θ through T as in Definition 3.1). Theorem 3.7. Take an arbitrary triple (X , Y, Z) of linear spaces. The linear spaces b[X ×Y, Z] and L[T, Z] are isomorphic. That is, with T = X ⊗ Y, b[X ×Y, Z] ∼ = L[X ⊗ Y, Z]. Proof. Take an arbitrary Φ ∈ L[T, Z]. The composition Φ ◦ θ : X ×Y → Z lies in b[X ×Y, Z] since θ is bilinear and Φ is linear (composition of a bilinear map with a linear transformation is a bilinear map). Then consider the transformation Cθ : L[T, Z] → b[X ×Y, Z] defined by Cθ (Φ) = Φ ◦ θ ∈ b[X ×Y, Z]

for every

Φ ∈ L[T, Z].

According to Corollary 3.4, for every φ in b[X ×Y, Z] there exists one and only one Φ in L[T, Z] for which φ = Φ ◦ θ. Then Cθ is injective and surjective, thus invertible. Moreover, Cθ is linear. In fact, Cθ (Φ1 + Φ2 ) = (Φ1 + Φ2 ) ◦ θ = (Φ1 ◦ θ) + (Φ2 ◦ θ) = Cθ (Φ1 ) + Cθ (Φ2 ), Cθ (α Φ) = (α Φ) ◦ θ = α(Φ ◦ θ) = α Cθ (Φ), for every Φ, Φ1 , Φ2 ∈ L[T, Z] and α ∈ F . Therefore Cθ is an isomorphism.



Remark 3.8. Two Relevant Consequences of Theorem 3.7. (a) A third proof of Theorem 2.5 (based on Theorem 1.1 and Theorem 3.7). Theorem 2.5 says that every bilinear map on the Cartesian product of linear manifolds of a pair of linear spaces can be extended to a bilinear map on the Cartesian product of the linear spaces. Remark 2.7(a) gave a second proof of Theorem 2.5, and it was also suggested after proving Theorem 2.5 that still another proof can be obtained via Theorem 3.7. Here it is. Let M and N be linear manifolds of linear spaces X and Y. Let φ : M×N → Z be a bilinear map into a linear space Z. Consider the isomorphism Cθ

40

3. Algebraic Tensor Product

of L[M ⊗ N , Z] onto b[M×N , Z] such that φ in b[M×N , Z] and Φ in L[M ⊗ N , Z] are uniquely related by φ = Cθ (Φ) = Φ ◦ θ, as in the proof of Theorem 3.7, where θ in b[M×N , M ⊗ N ] is the natural bilinear map associated with M ⊗ N . Since M ⊗ N is a linear manifold of the linear space  in L[X ⊗ Y, Z] such X ⊗ Y, extend Φ over X ⊗ Y using Theorem 1.1 to get Φ       that Φ|M⊗N = Φ. Set φ = Cθ(Φ) = Φ ◦ θ where θ in b[X ×Y, X ⊗ Y] is the natural bilinear map associated with X ⊗ Y. Since restrictions of bilinear maps to Cartesian products of linear manifolds are bilinear (Remark 2.1), and since there is a unique natural bilinear map associated with a tensor product space  M×N . Thus φ extends φ over X ×Y since (Corollary 3.6), then θ = θ|  M×N = (Φ  M×N = Φ  M×N ) = Φ|  ◦ θ)|  ◦ (θ|  M⊗N ◦ θ = Φ ◦ θ = φ. φ| (b) For a second consequence, take a tensor product (T, θ) of X and Y. Let T  = L[T, F ] be the algebraic dual of the tensor product space T. Set Z = F in Theorem 3.7 and get an isomorphic representation of the algebraic dual T  = (X ⊗ Y) of the tensor product space T = X ⊗ Y of X and Y, viz., (X ⊗ Y) ∼ = b[X ×Y, F ]

(∼ = L[X , Y  ] ∼ = L[Y, X  ] by Remark 2.7(b) ).

Next take the Cartesian product X  ×Y  of the dual spaces X  = L[X , F ] and Y  = L[Y, F ] of X and Y, and let (T , θ ) be a tensor product of X  and Y  . Thus applying Theorem 3.7, again with Z = F but now for the tensor product space T = X  ⊗ Y  of X  and Y  , we get (as above) (X  ⊗ Y  ) ∼ = b[X  ×Y  , F ]. Theorem 3.9. Let T = X ⊗ Y be the tensor product space of X and Y, let E and D be nonempty subsets of X and Y respectively, and set

E,D = x ⊗ y ∈ X ⊗ Y : x ∈ E and y ∈ D . (a) If span E = X and span D = Y, then spanE,D = X ⊗ Y. (b) If E and D are linearly independent, then E,D is linearly independent. Therefore if E is a Hamel basis for X and D is a Hamel basis for Y, then E,D is a Hamel basis for T = X ⊗ Y. Proof. (a) Take an arbitrary element  from X ⊗ Y (cf. Theorem 3.3), =

n i=1

xi ⊗ yi .

If span E = X and span D = Y then consider any expansion of each xi ∈ X in terms of vectors ei,j from E and any expansion of each yi ∈ Y in terms of vectors di,k from D. Since x ⊗ y = θ(x, y) for each pair (x, y) ∈ X ×Y, and since the map θ : X ×Y → X ⊗ Y is bilinear, we get

3.2 Further Properties of Tensor Product Spaces

=

n m i=1

j=1

βj ei,j ⊗

 k=1

41

n,m, γk di,k = βj γk (ei,j ⊗ di,k ). i,j,k

Thus  lies in spanE,D . Then X ⊗ Y ⊆ spanE,D . So spanE,D = X ⊗ Y.  (b) Let E  = {ej }m j=1 and D = {dk }k=1 be arbitrary nonempty finite linearly independent subsets of E and D respectively. Consider the linear manifolds

M = span E  = span {ej }m j=1 ⊆ X

and

N = span D = span {dk } k=1 ⊆ Y

spanned by them. Set Z = L[F m , F ], which is identified with the linear space the map φ : M×N → Z given for of all ×mmatrices of entries in F . Take  m every u = j=1 βj ej in M and every v = k=1 γk dk in N by   φ(u, v) = βj γk ∈ Z for j = 1, . . . , m and k = 1, . . . , ,   where βj γk is the ×m matrix whose entries are the products βj γk of the coefficients of the unique expansion of arbitrary vectors u ∈ M and v ∈ N in terms of the linearly independent sets E  and D. This φ : M×N → Z is bilinear. (Indeed, φ(u + u , v) = φ(u, v) + φ(u , v), φ(u, v + v  ) = φ(u, v) + φ(u, v  ), αφ(u, v) = φ(αu, v) = φ(u, αv) for every u, u ∈ M, v, v  ∈ N and α ∈ F ). So, according to axiom (b) in Definition 3.1, consider the linear transformation Φ : M ⊗ N → Z such that φ = Φ ◦ θ. Then φ(u, v) = Φ (θ(u, v)) = Φ(u ⊗ v) m  for every u = j=1 βj ej ∈ M and v = k=1 γk dk ∈ N . In particular, Φ(ej ⊗ dk ) = φ(ej , dk ) = Πj,k , where Πj,k ∈ Z is the ×m matrix whose entry at position j, k is 1 and all other entries are 0. These matrices form a linearly independent set in Z. (In fact, {Πj,k }m, j,k=1 is the canonical Hamel basis for Z.) Take any pair of integers k  , j  and suppose ej  ⊗ dk is a linear combination of the remaining single tensors {ej ⊗ dk }j,k∈I  with I  = {j, k = 1 to m, : j = j  , k = k  }, say  ej  ⊗ dk = δj,k (ej ⊗ dk ).  j,k∈I

Then, as Φ : M ⊗ N → Z is linear,  Πj  ,k = Φ(ej  ⊗ dk ) =

j,k∈I 

δj,k Φ(ej ⊗ dk ) =

 j,k∈I 

δj,k Πj,k ,

and so δj,k = 0 for every j, k ∈ I  since {Πj,k }m, j,k=1 is linearly independent in

Z. Thus {ej ⊗ dk }m, j,k=1 is linearly independent in E,D . In other words, E ,D = {x ⊗ y ∈ X ⊗ Y : x ∈ E  and y ∈ D }

is a finite linearly independent subset of E,D whenever E  and D are finite linearly independent subsets of E and D. Therefore if E and D are linearly independent subsets of X and Y, then every finite subset of each of them is

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3. Algebraic Tensor Product

trivially linearly independent, and so is every finite subset of E,D as we saw above. But if every finite subset of E,D is linearly independent, then so is  E,D (cf. Proposition 1.A in Section 1.4). Corollary 3.10.

dim(X ⊗ Y) = dim X · dim Y.

Proof. Let E and D be Hamel bases for X and Y. Thus E,D is a Hamel basis

for T by Theorem 3.9. Therefore E,D = x ⊗ y ∈ X ⊗ Y : x ∈ E and y ∈ D is in a one-to-one correspondence with the Cartesian product E×D. (In fact, if x ⊗ y ∈ E,D , then (i) x in E is unique for a fixed y in D as E is linearly independent and, (ii) y in D is unique for a fixed x in E as D is linearly independent — cf. Propositions 3.B and 3.C in Section 3.4.) As #(E×D) = #E · #D by definition of product of cardinal numbers (where # stands for cardinality; see, e.g., [52, Problem 1.30]), #E,D = #E · #D. So we get the claimed the identity (by definition of dimension).  Corollary 3.11. (a) Tensor product is commutative up to an isomorphism, X ⊗Y ∼ = Y ⊗ X. (b) Isomorphic linear spaces yield isomorphic tensor product spaces: X ∼ = X and Y  ∼ =Y

=⇒

X ⊗ Y ∼ = X ⊗ Y.

(c) The converse holds if X  and Y  are embedded in X and Y, respectively. X ⊗ Y ∼ =X ⊗Y

=⇒

X ∼ = X and Y  ∼ = Y.

Proof. These are straightforward consequences of the dimension identity of Corollary 3.10, since two linear spaces are isomorphic if and only if they have the same dimension (cf. Proposition 1.C in Section 1.4), and the dimension of an embedded linear space is clearly not greater than the dimension of the linear space it is embedded in.  Since we are identifying isomorphic tensor products after Corollary 3.6, we might write X ⊗ Y = Y ⊗ X and X  ⊗ Y  = X ⊗ Y in Corollary 3.11, but interpreting ∼ = as = in this particular case should perhaps be avoided. Remark 3.12. Isomorphism, Duality, and Finite-Dimensional Spaces. (a)

X ∼ = X , Y ∼ = Y , X ⊗ Y ∼ = (X ⊗ Y) =⇒ X ⊗ Y ∼ = X ⊗Y  ∼ = (X ⊗ Y) .

by Corollary 3.11(b). In this particular case Remark 3.8(b) ensures that X ⊗Y ∼ = X ⊗ Y ∼ = (X ⊗ Y) ∼ = b[X , Y, F ]. (b) If X and Y are finite-dimensional, then so are their algebraic duals X  and Y  , and a linear space and its dual have the same finite dimension. Moreover, dimensions coincide if and only if the linear spaces are isomorphic (cf. Propositions 1.C and 1.V in Section 1.4). Then

3.2 Further Properties of Tensor Product Spaces

43

dim X < ∞, dim Y < ∞ ⇒ dim X = dim X , dim Y = dim Y  ⇔ X ∼ = X , Y ∼ = Y . Thus Corollary 3.10 ensures that dim X ⊗ Y = dim X  ⊗ Y  = dim(X ⊗ Y) = dim(X  ⊗ Y  ) , or equivalently, X ⊗Y ∼ = (X ⊗ Y) ∼ = (X  ⊗ Y  ) . = X ⊗ Y ∼ Hence, according to Remark 3.8(b), dim X < ∞,

dim Y < ∞ =⇒

X ⊗Y ∼ = b[X ×Y, F ]. = (X ⊗ Y) ∼ = (X  ⊗ Y  ) ∼ = b[X  ×Y  , F ] ∼ = X ⊗ Y ∼ The following result identifies a special type of linear manifold of a tensor product space which will be referred to as a regular linear manifold . It shows that the tensor product space M ⊗ N of linear manifolds M and N of linear spaces X and Y, which is a linear manifold of X ⊗ Y, coincides with the linear manifold span R(θ|M×N ) of X ⊗ Y, so that M ⊗ N has as its natural bilinear map the restriction to M×N of the natural bilinear map θ of X ⊗ Y. Theorem 3.13. Let M and N be linear manifolds of linear spaces X and Y, respectively. Consider the tensor product (X ⊗ Y, θ) of X and Y, and take the tensor product space M ⊗ N . Then (a) M ⊗ N = span R(θ|M×N )

(i.e., M ⊗ N = span θ(M×N )),

(b) and so (M ⊗ N , θ ) is a tensor product of M and N with θ = θ|M×N . Proof. Let X and Y be linear spaces and take a tensor product (X ⊗ Y, θ). According to Definition 3.1 this means: X ⊗ Y is a linear space such that X ⊗ Y = span R(θ) for some bilinear map θ : X ×Y → R(θ) ⊆ X ⊗ Y with the property that for every bilinear map φ : X ×Y → Z into an arbitrary linear space Z there exists a linear transformation Φ : X ⊗ Y → Z such that φ = Φ ◦ θ. Now let M and N be linear manifolds of X and Y respectively. Since a restriction of a bilinear map to a Cartesian product of linear manifolds is bilinear (cf. Remark 2.1), then take the restriction θ|M×N of the natural bilinear map θ of X ⊗ Y to the Cartesian product M×N ⊆ X ×Y so that θ|M×N : M×N → R(θ|M×N ) ⊆ span R(θ|M×N ) ⊆ span R(θ) = X ⊗ Y is a bilinear map into the linear space span R(θ|M×N ), which is a linear manifold of the linear space X ⊗ Y = span R(θ). Take an arbitrary bilinear map φ : X ×Y → Z into an arbitrary linear space Z and consider its restriction φ|M×N to M×N , which is again a bilinear map. Thus

44

3. Algebraic Tensor Product

φ|M×N = (Φ ◦ θ)|M×N = Φ ◦ θ|M×N = Φ| spanR(θ|M×N ) ◦ θ|M×N , where Φ| spanR(θ|M×N ) : span R(θ|M×N ) → Z is a linear transformation, the restriction of the associated linear transformation Φ : X ⊗ Y → Z. By Theorem 2.5, every bilinear map on M×N has a bilinear extension over X ×Y. Then every bilinear map φ : M×N → Z is the restriction of a bilinear map φ : X ×Y → Z. Equivalently, every bilinear map φ : M×N → Z is of the form φ = φ|M×N : M×N → Z for some bilinear map φ : X ×Y → Z. Then, by the above displayed identity, for every bilinear map φ : M×N → Z there exists a linear transformation Φ| spanR(θ|M×N ) : span R(θ|M×N ) → Z such that φ = Φ| spanR(θ|M×N ) ◦ θ|M×N . So M ⊗ N = span R(θ|M×N ) by Definition 3.1, and hence θ = θ|M×N (the bilinear restriction of θ), is the natural bilinear map of M ⊗ N . Therefore (M ⊗ N , θ|M×N ) is a tensor product. 

This proves (a) and (b).

Definition 3.14. A linear manifold Υ of a tensor product space T = X ⊗ Y is regular if Υ = M ⊗ N for some linear manifolds M and N of the linear spaces X and Y. Otherwise Υ is called irregular . For a collection of properties of regular linear manifolds of tensor product spaces see [50, 51, 53]. Regular linear manifolds are characterized as follows. Corollary 3.15. A linear manifold Υ of T = X ⊗ Y is regular if and only if either it is zero or there exist Hamel bases E for X and D for Y such that Υ = spanE ,D , where E ,D = {x ⊗ y ∈ X ⊗ Y : x ∈ E  and y ∈ D } for some nonempty subsets E  ⊆ E and D ⊆ D. Proof. Suppose Υ = {0}. Let M and N be linear manifolds of X and Y. If Υ = M ⊗ N , then Υ = spanE ,D for some Hamel bases E  and D for M and N by Theorem 3.9, and so there are Hamel bases E and D for X and Y such that E  ⊆ E and D ⊆ D (see, e.g., [52, Theorem 2.5]). On the other hand, suppose Υ = spanE ,D with E  ⊆ E and D ⊆ D for Hamel bases E and D for X and Y. Then E  and D are Hamel bases for M = span E  ⊆ X and N = span D ⊆ Y. Therefore spanE ,D = M ⊗ N by Theorem 3.9. 

3.3 Tensor Product of Linear Transformations Now we focus on the concept of tensor product of linear transformations. Let X , Y, V, W be linear spaces, and let L[X , V] and L[Y, W] be the linear spaces

3.3 Tensor Product of Linear Transformations

45

of all linear transformations of X into V and of Y into W, respectively, all over the same field F . Consider the tensor products (X ⊗ Y, ϑ) of the linear spaces X and Y, and (V ⊗ W, ϑ ) of the linear spaces V and W. Definition 3.16. Let X , Y, V, W be linear spaces and consider the tensor product spaces X ⊗ Y and  V ⊗ W. Let A ∈ L[X , V] and B ∈ L[Y, W] be linear n transformations. For each i=1 xi ⊗ yi in X ⊗ Y set n n (A ⊗ B) xi ⊗ yi = Axi ⊗ Byi in V ⊗ W. i=1

i=1

It is shown below that this defines a linear transformation A ⊗ B of the linear space X ⊗ Y into the linear space V ⊗ W, referred to as the tensor product of the transformations A and B, or the tensor product transformation A ⊗ B. Theorem 3.17. Let X ⊗ Y and V ⊗ W be tensor product spaces of linear spaces X , Y, V, W. Take A ∈ L[X , V] and B ∈ L[Y, W]. (a) In fact, A ⊗ B in Definition 3.16 defines a linear transformation, A ⊗ B ∈ L[X ⊗ Y, V ⊗ W], and the value of A ⊗  B at  ∈ X ⊗ Y (i.e., (A ⊗ B)) does not depend n on the representation i=1 xi ⊗yi of . (b) The map θ : L[X , V]×L[Y, W] → L[X ⊗ Y, V ⊗ W] defined by θ(A, B) = A ⊗ B

for every

(A, B) ∈ L[X , V]×L[Y, W],

with A ⊗ B ∈ L[X ⊗ Y, V ⊗ W] as in (a), is bilinear . (c) Set L[X , V] ⊗ L[Y, W] = span R(θ). Then (L[X , V] ⊗ L[Y, W], θ) is a tensor product of the linear spaces L[X , V] and L[Y, W], and so L[X , V] ⊗ L[Y, W] ⊆ L[X ⊗ Y, V ⊗ W]. (d) The transformation A ⊗ B ∈ L[X ⊗ Y, V ⊗ W] in (a) coincides with a single tensor in the tensor product space L[X , V] ⊗ L[Y, W]: A ⊗ B ∈ L[X , V] ⊗ L[Y, W] ⊆ L[X ⊗ Y, V ⊗ W]. Proof. (a) Take the tensor product (X ⊗ Y, ϑ). Since A and B are linear transformations in L[X , Y] and L[V, W], the map ψ from the Cartesian product X ×Y to the linear space V ⊗ W given by ψ(x, y) = Ax ⊗ By for every (x, y) in X ×Y is bilinear. By Definition 3.1 and Corollary 3.4, there is a unique Ψ ∈ L[X ⊗ Y, V ⊗ W] such that ψ = Ψ ◦ ϑ; that is, such that the diagram ψ

X ×Y −−−→ V ⊗ W  ⏐  ⏐Ψ ϑ  X ⊗Y

46

3. Algebraic Tensor Product

n commutes. Take an arbitrary  = i=1 xi ⊗yi ∈ X ⊗Y. Thus by Definition 3.16, n n (A ⊗ B)() = Axi ⊗ Byi = ψ(xi , yi ) i=1 i=1 n n Ψ (ϑ(xi , yi )) = Ψ xi ⊗ yi = Ψ (). = i=1

i=1

Then A ⊗ B = Ψ. So A ⊗ B is linear and (A ⊗ B)() does not depend on the representation of  as Ψ is linear and does not depend on the representation of . n (b) Take A, A ∈ L[X , V], B, B  ∈ L[Y, W],  = i=1 xi ⊗ yi ∈ X ⊗ Y, and also (b) we get θ(A + A , B) = α ∈ F , all arbitrary. For nthe map θ defined in item n   ) ⊗ B)  = (A + A )x ⊗ By = ((A + A i i i=1 i=1 (Ax  in+ A x i ) ⊗ Byi = n n  (Ax ⊗ By ) + (A x ⊗ By ) = (Ax ⊗ By ) + (A xi ⊗ Byi ) = i i i i i i i=1 i=1 i=1 (A ⊗ B) + (A ⊗ B) = θ(A, B) + θ(A , B). Since this holds for every , θ(A+A , B) = θ(A, B)+ θ(A , B). Similarly, θ(A, B +B  ) = θ(A, B)+ θ(A, B  ) and α θ(A, B) = θ(α A, B) = θ(A, α B). Therefore θ : L[X , V]×L[Y, W] → R(θ) ⊆ span R(θ) ⊆ L[X ⊗ Y, V ⊗ W] is bilinear. (c) According to the definition of θ,

span R(θ) = span A ⊗ B ∈ L[X ⊗ Y, V ⊗ W] : A ∈ L[X , V], B ∈ L[Y, W] n

= j=1 Aj ⊗ Bj ∈ L[X ⊗ Y, V ⊗ W] : A ∈ L[X , V], B ∈ L[Y, W], n ∈ N . If Z is an arbitrary linear space over F , and if φ : L[X , V]×L[Y, W] → Z is an arbitrary bilinear map on L[X , V]×L[Y, W] with values in Z, then associate with φ a transformation Φ : span R(θ) → Z defined by  n n Aj ⊗ Bj = φ(Aj , Bj ) ∈ Z Φ j=1 j=1 n for every j=1 Aj ⊗ Bj ∈ span R(θ) ⊆ L[X ⊗ Y, V ⊗ W]. Since φ is a bilinear map, then it is readily verified that Φ is a linear transformation, Φ ∈ L[span R(θ), Z].    Aj ⊗ Bj and k Ak ⊗ Bk in L[X⊗ Y, V ⊗ W] it (Indeed, for anyelements j    = j φ(Aj , Bj ) + k φ(Ak , Bk ) = follows j ⊗ Bj + k Ak ⊗ Bk )  that Φ( j A   Φ(  j Aj ⊗ Bj ) = Φ( j αAj ⊗ Bj ) =  j Aj ⊗ Bj )+Φ( k Ak ⊗ Bk ), and Φ(α j φ(αAj , Bj ) = α j φ(Aj , Bj ) = α Φ( j Aj ⊗ Bj ) for α ∈ F . So Φ is linear.) Furthermore, for every (A, B) ∈ L[X , V]×L[Y, W],   (Φ ◦ θ)(A, B) = Φ θ(A, B) = Φ(A ⊗ B) = φ(A, B). Thus φ = Φ ◦ θ; equivalently, the diagram φ

L[X , V]×L[Y, W]

−−−→ θ





Z  ⏐ ⏐Φ

span R(θ) commutes. Hence (span R(θ), θ) satisfies the axioms of Definition 3.1, and therefore (L[X , V] ⊗ L[Y, W], θ) is the tensor product of L[X , V] and L[Y, W].

3.3 Tensor Product of Linear Transformations

47

(d) A single tensor is by definition the value of the natural bilinear map θ at a pair. Thus by definition of θ the value of it at a pair (A, B) is precisely the tensor product transformation A ⊗ B in (a).  Remark 3.18. Tensor Product of Duals and Dual of Tensor Products. (a) It was proved in Theorem 3.17(a) that A ⊗ B, defined in Definition 3.16, is a function on X ⊗ Y, and as such (A ⊗ B) cannot depend on the representation of  in X ⊗ Y (which is the case for every transformation L ∈ L[X ⊗ Y, Z] into any linear space Z). There are two particular cases worth noticing. (i) V = X , W = Y =⇒ A ∈ L[X ], B ∈ L[Y] and A ⊗ B ∈ L[X ⊗ Y]. (ii) V = W = F =⇒ L[X , V] = L[X , F ] = X  , L[Y, W] = L[Y, F ] = Y  and L[X ⊗ Y, V ⊗ W] = L[X ⊗ Y, F ⊗ F ] = L[X ⊗ Y, F ] = (X ⊗ Y) . Case (i) is a trivial particularization; case (ii) deserves a few words. (b) Consider the tensor product (V ⊗ W, ϑ ) with V = W = F . First note by Corollary 3.10 that dim(F ⊗ F ) = dim F = 1, thus F ⊗ F ∼ = F and, as usual, identification with a field will be interpreted as identity and we write F ⊗ F = F , which justifies writing L[X ⊗ Y, F ⊗ F ] = L[X ⊗ Y, F ] = (X ⊗Y) . In this case the natural bilinear map ϑ : F ×F = F 2 → F ⊗ F = F is given by ϑ (α, β) = α ⊗ β = α β

for every

α, β ∈ F .

So for every f ∈ X  and g ∈ Y  , n n f (xi ) ⊗ g(yi ) = f (xi ) g(yi ) ∈ F ⊗ F = F for  ∈ X ⊗ Y (f ⊗ g) = i=1

i=1

and, in particular, (f ⊗ g)(x ⊗ y) = f (x) g(y)

for every

x ∈ X and y ∈ Y,

where f ⊗ g is a tensor product transformation in L[X ⊗ Y, F ⊗ F ] = (X ⊗ Y), and a single tensor in the tensor product space L[X , F ] ⊗ L[Y, F ] = X  ⊗ Y  . Thus, since L[X , F ] ⊗ L[Y, F ] ⊆ L[X ⊗ Y, F ⊗ F ] by Theorem 3.17(c), X  ⊗ Y  ⊆ (X ⊗ Y) . Theorem 3.19. Let V, W, X , Y, X , Y  be linear spaces over the same field F . If α, β ∈ F , A, A1 , A2 ∈ L[X , V], B, B1 , B2 ∈ L[Y, W], C ∈ L[X , X ] and D ∈ L[Y , Y], then the following identities hold true. (a) α β (A ⊗ B) = αA ⊗ βB = (α βA) ⊗ B = A ⊗ (α βB). (b) (A1 + A2 ) ⊗ (B1 + B2 ) = A1 ⊗ B1 + A2 ⊗ B1 + A1 ⊗ B2 + A2 ⊗ B2 . (c) A C ⊗ BD = (A ⊗ B) (C ⊗ D).

48

3. Algebraic Tensor Product

(d) A,B are invertible if and only if A ⊗ B is, and (A ⊗ B)−1 = A−1 ⊗ B −1 . (e) R(A) ⊗ R(B) = R(A ⊗ B). (f) A ⊗ B is surjective if and only if both A and B are surjective: R(A ⊗ B) = V ⊗ W ⇐⇒ R(A) = V and R(B) = W . (g) A ⊗ B is injective if and only if both A and B are injective: N (A ⊗ B) = {0} ⇐⇒ N (A) = {0} and N (B) = {0} .     (h) N (A) ⊗ N (B)  N (A ⊗ B) = N (A) ⊗ Y + X ⊗ N (B) . (i) A ⊗ B ∼ = B ⊗ A. (j) (A ⊗ B) = A ⊗ B 

(as transformations in L[(V ⊗ W) , (X ⊗ Y) ]).

Proof. If a linear space is null, then the above results are trivial or meaningless. So suppose they are nonzero. Recall: when dealing with tensor products, identities are interpreted up to isomorphisms, and inclusions mean embeddings. (a,b) The results in (a) and (b) are straightforward since A ⊗ B is a single tensor in the tensor product space L[X , V] ⊗ L[Y, W] and so A ⊗ B is the value of a bilinear map θ at the pair (A, B) in L[X , V]×L[Y, W]. (c,d) Take A ∈ L[X , V], B ∈ L[Y, W], C ∈ L[X , X ], and D ∈ L[Y , Y], consider the tensor product transformations A ⊗ B in L[X  ⊗ Y, V ⊗ W] and C ⊗ D in L[X  ⊗ Y , X ⊗ Y], and take an arbitrary  = i vi ⊗ wi in V ⊗ W. Then 

 

(A ⊗ B) (C ⊗ D)  = (A ⊗ B) (C ⊗ D) = (A ⊗ B) Cvi ⊗ Dwi i  = A Cvi ⊗ BDwi = (A C ⊗ BD), i

which proves (c). Setting X  = V, Y  = W, C = A−1 in L[V, X ] and D = B −1 in L[W, Y], then (c) implies (d): (A ⊗ B) (A−1 ⊗ B −1 ) = IV ⊗ IW

and

(A−1 ⊗ B −1 ) (A ⊗ B) = IX ⊗ IY ,

where IX , IY , IV and IW stand for the identities on X , Y, V and W, and IX ⊗ IY and IV ⊗ IW are the identities on X ⊗ Y and V ⊗ W. (e,f) An B) for some  element G lies in R(A ⊗ B) if and only  if G = (A ⊗  x ⊗ y ∈ X ⊗ Y, which means G = Ax ⊗ By = i i i i i i i ui ⊗ vi for ui ∈ R(A) and vi ∈ R(B) or, equivalently, G ∈ R(A) ⊗ R(B). So we get (e):

 =

R(A ⊗ B) = R(A) ⊗ R(B). On the other hand (e) implies (f). In fact, since R(A) ⊗ R(B) ⊆ V ⊗ W, then R(A) = V and R(B) = W implies R(A) ⊗ R(B) = V ⊗ W by (e), and the converse is straightforward by (e) and Corollary 3.11(c). This proves (f). (g) Suppose one of A or B is not injective. With no loss of generality, suppose A is not injective (i.e., N (A) = {0}) and take 0 = x ∈ N (A) (i.e., Ax = 0 and

3.3 Tensor Product of Linear Transformations

49

x = 0) so that (A ⊗ B)(x ⊗ y) = Ax ⊗ By = 0 for an arbitrary 0 = y ∈ Y. As x ⊗ y = 0 (cf. Proposition 3.C), then there exists x ⊗ y = 0 in N (A ⊗ B). So A ⊗ B is not injective (i.e., N (A ⊗ B) = {0}). Equivalently, if A ⊗ B is injective, then so are both A and B. For the converse, suppose  A and B are  = injective. If A ⊗ B is not  injective, then there exists i xi ⊗ yi = 0 in  X ⊗ Y such that (A ⊗ B) i xi ⊗ yi = i Axi ⊗ Byi = 0. Thus {Axi ⊗ Byi } is not linearly independent, and so one of {Axi } or {Byi } is not linearly independent (cf. Theorem 3.9). With no loss of generality suppose {Axi } is not linearly independent. But we can take a representation of the same  with {xi } being linearly independent (cf. Proposition 3.A). Hence {Axi } is linearly independent because A is injective. This leads to a contradiction. So if A and B are injective, then A ⊗ B is injective. This proves (g). (h) Take the  linear manifold N (A) ⊗ N (B) of X ⊗ Y. If  ∈ N (A) ⊗ N (B), then  = i xi ⊗ yi with xi in N (A)  and yi in N (B), and so Axi = 0 and Byi = 0, which implies (A ⊗ B) = i Axi ⊗ Byi = 0. Thus  ∈ N (A ⊗ B). Hence N (A) ⊗ N (B) ⊆ N (A ⊗ B) for every A, B. The reverse inclusion fails if one of A or B is not injective and the other is not null. Indeed, take x ∈ N (A) and 0 = y ∈ N (B) so that 0 = x ⊗ y ∈ N (A) ⊗ N (B) but x ⊗ y ∈ N (A ⊗ B) as (A ⊗ B)(x ⊗ y) = Ax ⊗ By = Ax ⊗ 0 = 0. So there are A and B for which N (A ⊗ B) ⊆ N (A) ⊗ N (B). To prove the identity N (A ⊗ B) = N (A) ⊗ Y + X ⊗ N (B), suppose A and B are not null (otherwise the identity is trivial). Let M and N be the nonzero algebraic complements of N (A) and N (B) in X and Y, that is, X = N (A) + M

and

Y = N (B) + N .

Then A|M ∈ L[M, V] and B|N ∈ L[N , W] areinjective (i.e., N (A|M ) = {0} and N (B|N ) = {0}). Take an arbitrary  = i xi ⊗ yi ∈ X ⊗ Y. Write xi = ai + ui and yi = bi + vi with ai ∈ N (A), ui ∈ M, bi ∈ N (B), vi ∈ N so that xi ⊗ yi = (ai + ui ) ⊗ (bi + vi ) = (ai ⊗ bi ) + (ai ⊗ vi ) + (ui ⊗ bi ) + (ui ⊗ vi ), and therefore (since Aai = 0 and Bbi = 0)     (A ⊗ B) = (A ⊗ B) a ⊗ b + a ⊗ v + u ⊗ b + u ⊗ v i i i i i i i i i i i i     = i Aai ⊗ Bbi + i Aai ⊗ Bvi + i Aui ⊗ Bbi + i Aui ⊗ Bvi   = i Aui ⊗ Bvi = i A|M ui ⊗ B|N vi .  Suppose  ∈ N  = 0 and hence i A|M ui ⊗ B|N vi = (A ⊗ B). Thus (A ⊗ B) (A|M ⊗ B|N ) i ui ⊗ vi = 0 so that i ui ⊗ vi ∈ N (A|M ⊗ B|N ). But the restrictions A|M ∈ L[M, V] and B|N ∈ L[N , W] are injective (by their definition according to the decomposition X = N (A) + M and Y = N (B) + N ), and so A|M ⊗ B|N is injective (i.e., N (A|M ⊗ B|N ) = {0}) by (g). Hence  ui ⊗ vi = 0. i

50

3. Algebraic Tensor Product

Thus      = i xi ⊗ yi = i ai ⊗ bi + i ai ⊗ vi + i ui ⊗ bi   = i ai ⊗ (bi + vi ) + i ui ⊗ bi ⊆ N (A) ⊗ Y + X ⊗ N . Then N (A ⊗ B) ⊆ N (A) ⊗ Y + X ⊗ N (B). The reverse inclusion is straightforward. Indeed, if  ∈ N (A) ⊗ Y + X ⊗ N (B) ⊆ X ⊗ Y, then (A ⊗ B) ⊆ A(N (A)) ⊗ B(Y) + A(X ) ⊗ B(N (B)) = {0} ⊗ R(B) + R(A) ⊗ {0} = {0}, and so  ∈ N (A ⊗ B). Thus N (A) ⊗ Y + X ⊗ N (B) ⊆ N (A ⊗ B). Therefore N (A) ⊗ Y + X ⊗ N (B) = N (A ⊗ B). ∼ Y ⊗ X and V ⊗ W ∼ (i) By Corollary 3.11(a), X ⊗ Y = = W ⊗ V. It is readily : X ⊗ Y → Y ⊗ X and Π : V ⊗ W → W ⊗ V given by Π1  = verified that Π 1 2    y ⊗ x and Π G = w ⊗ v for every  = x ⊗ yi ∈ X ⊗ Y and G = i 2 v i i i i i i v ⊗ w ∈ V ⊗ W are invertible linear transformations i i i (called the natural permutation isomorphisms). Thus for an arbitrary  = i xi ⊗ yi ∈ X ⊗ Y,   Π2 (A ⊗ B) = Π2 i Axi ⊗ Byi = i Byi ⊗ Axi   = (B ⊗ A) i yi ⊗ xi = (B ⊗ A)Π1 i xi ⊗ yi = (B ⊗ A)Π1 . Hence Π2 (A ⊗ B) = (B ⊗ A)Π1 . That is, tensor product of linear transformations is commutative up to isomorphisms, which means A ⊗ B and B ⊗ A are isomorphically equivalent: A ⊗ B ∼ = B ⊗ A. In particular, if V = X and W = Y so that A ∈ L[X ] and B ∈ L[Y], then A ⊗ B and B ⊗ A are similar. (j) Take A in L[X , V], B in L[Y, W], A in L[V  , X  ], B  in L[W  , Y  ], A ⊗ B in L[X , V] ⊗ L[Y, W] ⊆ L[X ⊗ Y, V ⊗ W], its algebraic adjoint (A ⊗ B) in L[(V ⊗ W) , (X ⊗ Y) ], and the tensor product of algebraic adjoints A ⊗ B  in L[V , X  ] ⊗ L[W  , Y  ] ⊆ L[V  ⊗ W  , X  ⊗ Y  ] ⊆ L[(V ⊗ W) , (X ⊗ Y) ]. Take any single tensor f ⊗ g ∈ V  ⊗ W  ⊆ (V ⊗ W) . By definition of tensor product transformation and of algebraic adjoint, (A ⊗ B  )(f ⊗ g) = A f ⊗ B  g = fA ⊗ gB ∈ X  ⊗ Y  ⊆ (X ⊗ Y) , (A ⊗ B) (f ⊗ g) = (f ⊗ g)(A ⊗ B) ∈ (X ⊗ Y) . Cliam.

(∗) (∗∗)

(f ⊗ g)(A ⊗ B) = fA ⊗ gB ∈ X  ⊗ Y  ⊆ (X ⊗ Y) .

Proof of Claim. Take any single tensor x ⊗ y in X ⊗ Y, and an arbitrary single tensor A ⊗ B in L[X , V] ⊗ L[Y, W]. By definition of tensor product transformation, (A ⊗ B)(x ⊗ y) = Ax ⊗ By, again a single tensor now in V ⊗ W. Next take an arbitrary single tensor f ⊗ g in V  ⊗ W  = L[V, F ] ⊗ L[W, F ] so that, by definition of tensor product transformation again, (f ⊗ g)(Ax ⊗ By) = f Ax ⊗ gBy ∈ F ⊗ F = F . On the other hand, since f A ∈ X  = L[X , F ] and gB ∈ Y  = L[Y, F ], we get by the definition of tensor product transformation once again that (f A ⊗ gB)(x ⊗ y) = f Ax ⊗ gBy ∈ F ⊗ F = F . Summing up:

3.3 Tensor Product of Linear Transformations

51

(f ⊗ g)(A ⊗ B)(x ⊗ y) = (f ⊗ g)(Ax ⊗ By) = fAx ⊗ gBy = (fA ⊗ gB)(x ⊗ y)  for i xi ⊗ yi =  every x ⊗ y ∈ X ⊗ Y. So (f⊗ g)(A ⊗ B) = (f ⊗ g)(A ⊗ B)  (f ⊗ g)(A ⊗ B)(x ⊗ y ) = (fA ⊗ gB)(x ⊗ y ) = (fA ⊗ gB) i i i i i i i xi ⊗ yi = (fA ⊗ gB) for every  ∈ X ⊗ Y, and hence (f ⊗ g)(A ⊗ B) = fA ⊗ gB in X  ⊗ Y  = L[X , F ] ⊗ L[Y, F ] ⊆ L[X ⊗ Y, F ] = (X ⊗ Y) .



Then by (∗), (∗∗) and the above claim for every f ⊗ g ∈ V  ⊗ W  ⊆ (V ⊗ W) .  Thus for an arbitrary finite sum Ω = i fi ⊗ gi in V  ⊗ W  ⊆ (V ⊗ W) we get (A ⊗ B  )Ω = (A ⊗ B) Ω for every Ω ∈ V  ⊗ W  . Hence

(A ⊗ B  )(f ⊗ g) = (A ⊗ B) (f ⊗ g)

A ⊗ B  = (A ⊗ B)

on

V  ⊗ W  ⊆ (V ⊗ W) ,

in L[V  , X  ] ⊗ L[W  , Y  ] ⊆ L[V  ⊗ W  , X  ⊗ Y  ] ⊆ L[(V ⊗ W) , (X ⊗ Y) ]. This concludes the proof of (j) and, consequently, of Theorem 3.19.  Remark 3.20. Basic Properties and Tensor Sums of Transformations. (a) Take A ∈ L[X , V] and B ∈ L[Y, W] arbitrarily. Take any 0 = α ∈ F . Thus A ⊗ B = αA ⊗ α1 B by Theorem 3.19(a). Similarly, using the same notation for the null transformation in the linear spaces L[X , V] and L[Y, W], it follows that the null transformation in L[X , V] ⊗ L[Y, W] ⊆ L[X ⊗ Y, V ⊗ W] can be written as A ⊗ O = O ⊗ B = O ⊗ O. Products are not uniquely represented, and so single tensors do not have a unique representation (as we saw before). This tautologically extends to transformations (as we saw above): a tensor product of linear transformations does not have a unique representation. However, as happens with products in general (and with every sort of single tensors in particular), A = O and A ⊗ B1 = A ⊗ B2

=⇒

B1 = B2

by Proposition 3.C in Section 3.4. Symmetrically, B = O and A1 ⊗ B = A2 ⊗ B

=⇒

A1 = A2 .

If X = V and Y = W and if IX : X → X and IY : Y → Y stand for the identity transformations on X and on Y, then the identity I : X ⊗ Y → X ⊗ Y on X ⊗ Y is such that for any 0 = α ∈ F I = IX ⊗ IY = αIX ⊗ α1 IY . Also, according to Theorem 3.19(c), for every A ∈ L[X ] and B ∈ L[Y] we get

52

3. Algebraic Tensor Product

A ⊗ B = (A ⊗ IY ) (IX ⊗ B) = (IX ⊗ B) (A ⊗ IY ). (b) Motivated by the above identity, the tensor sum of A ∈ L[X ] and B ∈ L[Y] is defined as the transformation A  B : X ⊗ Y → X ⊗ Y given by A  B = (A ⊗ IY ) + (IX ⊗ B), which lies in L[X ⊗ Y]. (It is sometimes written ⊕ instead of , but we reserve the symbol ⊕ for direct sum, as usual.) It is worth noticing that the tensor sum is not commutative even if X = Y with O, A, B, I lying in L[X ]. Indeed, A  O = A ⊗ I, A  I = A ⊗ I + I ⊗ I,

O  B = I ⊗ B, I  B = I ⊗ I + I ⊗ B.

In particular, if A = B, then A  O = A ⊗ I = I ⊗ A = O  A, A  I = A ⊗ I + I ⊗ I = I ⊗ I + I ⊗ A = I  A.

3.4 Additional Propositions All linear  spaces are nonzero and over the same field F , all sums of single tensors i xi ⊗ yi are finite, and ∼ = stands for isomorphism. As agreed at the end of Section 3.1, and according to Corollary 3.6, equality of tensor products is regarded up to an isomorphism and so the equality or non-equality signs referring to tensor product spaces may be interpreted as ∼  . = or ∼ = Proposition 3.A. Let  be an arbitrary element in the tensor product space T = X ⊗ Y of  linear spaces X and Y. There exists a representation for  such that  = i xi ⊗ yi with linearly independent sets {xi } and {yi }. n Proposition 3.B. Let i=1 xi ⊗ yi be a representation with n single tensors for an arbitrary element  in the tensor product space T = X ⊗ Y of linear spaces X and Y. If the set {xi }ni=1 is linearly independent, nthen the set n } is uniquely determined by  in the following sense: if {y i=1 xi ⊗ yi = in i=1   x ⊗ y , then y = y for every i = 1, . . . , n. Symmetrically, if the set i i i i=1 i {yi }ni=1 is linearly independent, then {xi }ni=1 is uniquely determined by . Proposition 3.C. Consider the above setup. Suppose x, u ∈ X and y, v ∈ Y. (a) x ⊗ y = 0 if and only if one of x or y is zero. (b) x ⊗ y = u ⊗ v = 0 if and only if u = αx and v =

1 αy

for some 0 = α ∈ F .

Proposition 3.D. For every triple (X , Y, Z) of linear spaces, (a) X ⊗ (Y ⊕ Z) ∼ = (X ⊗ Y) ⊕ (X ⊗ Z), ∼ (b) (X ⊕ Y) ⊗ Z = (X ⊗ Z) ⊕ (Y ⊗ Z), (c) (X ⊗ Y) ⊗ Z ∼ = X ⊗ (Y ⊗ Z).

3.4 Additional Propositions

53

Proposition 3.E. The following assertions hold for arbitrary linear manifolds M, M1 , M2 of a linear space X and N , N1 , N2 of a linear space Y. (a) (M1 ⊗ Y) ∩ (M2 ⊗ Y) = (M1 ∩ M2 ) ⊗ Y. (b) (X ⊗ N1 ) ∩ (X ⊗ N2 ) = X ⊗ (N1 ∩ N2 ). (c) (M ⊗ Y) ∩ (X ⊗ N ) = M ⊗ N . (d) (M1 ⊗ N1 ) ∩ (M2 ⊗ N2 ) = (M1 ∩ M2 ) ⊗ (N1 ∩ N2 ). Proposition 3.F. If X and Y are linear spaces over F and m, n ∈ N , then ∼ X n and F m ⊗ Y ∼ X ⊗ Fn = = Y m , and so F m ⊗ F n ∼ = F mn . In particular, X ⊗ F ∼ = X and F ⊗ Y ∼ = Y, and so F ⊗ F ∼ = F. Proposition 3.G. In general, for linear manifolds M and N of X and Y, (X /M) ⊗ (Y/N ) ∼  (X ⊗ Y)/(M ⊗ N ). = Proposition 3.H. If A ∈ L[X , V] and B ∈ L[Y, W], then (X /N (A)) ⊗ (Y/N (B)) ∼ = (X ⊗ Y)/N (A ⊗ B). Proposition 3.I. If M and N are linear manifolds of X and Y, then M ⊗ N = X ⊗ Y ⇐⇒ M = X and N = Y. The converse of Corollary 3.10 reads as follows. Proposition 3.J. Let X , Y, T be linear spaces over the same field and take a bilinear map θ : X ×Y → T. If dim T = dim X · dim Y and span θ(X ×Y) = T, then (T, θ) is a tensor product of X and Y. Proposition 3.K. Let (T, θ) be a tensor product of linear spaces X and Y and let Υ ⊆ T be a linear manifold of the tensor product space T = X ⊗ Y. If Υ is regular, then (Υ, θ ) is a tensor product of linear manifolds M and N of X and Y such that Υ is isomorphic to span θ(M×N ). Moreover, there is a unique isomorphism Θ : span θ(M×N ) → Υ for which θ = Θ ◦ θ|M×N . The next result is the converse of Theorem 3.19(d). Proposition 3.L. Consider the setup of Theorem 3.19. Suppose A ⊗ B ∈ L[X , V] ⊗ L[Y, W] ⊆ L[X ⊗ Y, V ⊗ W] is invertible. If its inverse (A ⊗ B)−1 ∈ L[V ⊗ W, X ⊗ Y] is such that (A ⊗ B)−1 = C ⊗ D ∈ L[V, X ] ⊗ L[W, Y] ⊆ L[V ⊗ W, X ⊗ Y] for some C ∈ L[V, X ] and D ∈ L[W, Y] with one of them invertible, then A ∈ L[X , V] and B ∈ L[Y, W] are invertible and (A ⊗ B)−1 = A−1 ⊗ B −1 . Some additional properties related to Theorem 3.19(h) are given below.

54

3. Algebraic Tensor Product

Proposition 3.M. Consider the setup of Theorem 3.19. Take λ, μ ∈ F . Let I stand for the identity in L[X ], in L[Y] and in L[X ⊗ Y]. The following holds.    (a) N (A) ⊗ Y ∪ (X ⊗ N (B)  N (A ⊗ B). (b) If dim X = ∞ and A is not injective, or if dim Y = ∞ and B is not injective, then dim N (A ⊗ B) = ∞. (c) N (λI − A) ⊗ N (μI − B) ⊆ N (λμI − A ⊗ B). The inclusions in Proposition the inclusion in Theorem  3.M(a,c) extend   3.19(h) since N (A) ⊗ N (B) ⊆ N (A) ⊗ Y ∩ X ⊗ N (B) . Basic operations with the tensor sums of Remark 3.20 are listed below. Proposition 3.N. For every α, β ∈ F , A, A1 , A2 ∈ L[X ] and B, B1 , B2 ∈ L[Y], (a) (α + β)(A  B) = αA  βB + βA  αB, (b) (A1 + A2 )  (B1 + B2 ) = A1  B1 + A2  B2 , (c) (A1  B1 )(A2  B2 ) = A1 ⊗ B2 + A2 ⊗ B1 + A1 A2  B1 B2 , (d) (A  B) = A  B  (as transformations in L[(X ⊗ Y) ]). Notes: An organized exposition of tensor products in book form dates back to Schatten’s 1950 monograph [85], where the notion of direct products of linear spaces was given in terms of formal products (which match what we call tensor product space and single tensors). This came after a Kronecker-product-like notion on finite-dimensional spaces given in Weyl’s 1931book [93, Chapter V]. Definition 3.1 provides an axiomatic starting point to introduce tensor products. This leads to the abstract notion of algebraic tensor products of linear spaces, where the concrete standard forms are shown to be interpretations of the axiomatic formulation. Such concrete standard forms (namely, the quotient space and the linear maps of bilinear maps formulations) will be the subject of the next chapter. A similar abstract presentation has been considered, for instance, in [66, Section XVI.1] and [81, Chapter 14] for the quotient space formulation, in [94, Section I.4] and [27, Chapter 1] for both formulations, and in [42, Section 1.6] and [15, Section 2.2] for the linear maps of bilinear maps formulation. See also the recent expository paper in [57]. The above propositions are basic properties resulting from such axiomatic presentation. Proposition 3.A is a useful property whose proof goes as follows. n Proof of Proposition 3.A. Take  = i=1 xi ⊗ yi ∈ X ⊗ Y with n > 1 to avoid trivialities. Consider the subsets {xi }ni=1 and {yi }ni=1 of X and Y, respectively, n then such that xk = (a)   there exists xk   If {xi }i=1 is not linearly independent, α x . So  = x ⊗ y + x ⊗ y = α x ⊗ y + i i k k i i i i k i=k i=k i=k xi ⊗ yi =  i=k  x ⊗ α y + x ⊗ y = x ⊗ (α y + y ). Set y = (αi yk + yi ) i i k i i i i k i i i=k i=k i=k  so that  = i=k xi ⊗ yi is a representation of  with a smaller number of

3.4 Additional Propositions

55

single tensors (with n −1 single tensors), where {xi }i=k is a proper subset of {xi }ni=1 . If {xi }i=k is still linearly dependent, then repeat the same procedure to get another representation for the same  with an even smaller number of single tensors in terms of a proper subset of {xi }i=k . Since the representation of  has a finite number of single tensors, this process will eventually reach a representation with a minimum number of terms in xi , and so the remaining subset of {xi }ni=1 must be linearly independent. (b) Now suppose {xi }m i=1 is already linearly independent. Apply the same prountil reaching a representation of  with a linearly indepencess to {yi }m i=1 . Since this finite process involves only successive linear dent subset of {yi }m i=1 combinations of the already linearly independent set {xi }m i=1 , and since sets obtained by taking linear combinations of subsets of a linearly independent set remain  linearly independent, then the resulting minimal representation of  = j xj ⊗ yj consists of a minimum number of single tensors xj ⊗ yj obtained from finite linearly independent sets {xj } and {yj }.  For Proposition 3.B, see, e.g., [94, Lemma 1.1, p. 9], which leads to Proposition 3.C (see, e.g., [5, Problem 2.U]). Proposition 3.D presents standard associative and distributive properties (see, e.g., [94, Exercise Section 5, p. 11 and Proposition 1.8, p. 13]). The intersections of regular subspaces in Proposition 3.E can be found in, for instance, [27, Section 1.15]. Proposition 3.F is an application of Corollary 3.10 and Proposition 1.C (since dim X n = n dim X in light of Proposition 1.T). To verify Proposition 3.G set X = F and M = {0}, and apply Proposition 3.F. Proposition 3.H can be verified by using the First Isomorphism Theorem (Theorem 1.4(b)) and Theorem 3.19(e) on the range of a tensor product transformation. Proposition 3.I is straightforward from Corollary 3.11(c). For Proposition 3.J see, e.g., [94, Lemma 1.2, p. 19]. Proof of Proposition 3.K. Let X , Y, Z be nonzero linear spaces over the same field. Let (T, θ) be a tensor product of X and Y. By Definition 3.1, for every bilinear map φ : X ×Y → Z there is a linear transformation Φ : T → Z such that φ = Φ ◦ θ. If Υ is a regular linear manifold of T , then (cf. Definition 3.14) (Υ, θ ) is a tensor product of linear manifolds M and N of X and Y. Take any bilinear map φ : M×N → Z. By Theorem 2.5, there is a bilinear extension φ : X ×Y → Z of φ over X ×Y. Thus φ = φ|M×N = (Φ ◦ θ)|M×N = Φ ◦ θ|M×N = Φ span θ(M×N ) ◦ θ|M×N , where Φ| span θ(M×N ) : span θ(M×N ) → Z is a linear transformation (the restriction of a linear transformation to a linear manifold is linear) associated with the bilinear map θ|M×N : M×N → Υ (the restriction of a bilinear map to the Cartesian product of linear manifolds is bilinear). So (cf. Definition 3.1) (span θ(M×N ), θ|M×N ) is a tensor product of M and N . Thus (Υ, θ ) and (span θ(M×N ), θ|M×N ) are tensor products of the same pair of linear spaces M and N , and so Theorem 3.5 says that there is a unique isomorphism Θ ∈ L[span θ(M×N ), Υ ] for which θ = Θ ◦ θ|M×N ). 

56

3. Algebraic Tensor Product

Proof of Proposition 3.L. If there is (A ⊗ B)−1 ⊆ L[V ⊗ W, X ⊗ Y] such that (A ⊗ B)−1 (A ⊗ B) = IX ⊗Y

and

(A ⊗ B)(A ⊗ B)−1 = IV⊗W ,

where IX ⊗Y = IX ⊗ IY and IV⊗W = IV ⊗ IW . These are the identities on X ⊗ Y, on V ⊗ W, and on X , Y, V, W, respectively. If, in addition, (A ⊗ B)−1 = C ⊗ D with C ∈ L[V, X ] and D ∈ L[W, Y], so that C ⊗ D = 0, then by Theorem 3.19(c) IX ⊗IY = (C⊗D)(A⊗B) = CA⊗DB,

IV ⊗IW = (A⊗B)(C⊗D) = AC⊗BD.

Thus Proposition 3.C(b) ensures that there are nonzero α, β ∈ F for which CA = α IX ,

DB =

1 α IY ,

and

AC = βIV , BD =

1 β IW .

If one of C or D is invertible, say, if C is invertible, then A = α C −1 = β C −1 . Hence (i) β = α and (ii) A is invertible with A−1 = α1 C. Since β = α we get (α D)B = IY and B(α D) = IW so that B is invertible with B −1 = α D. As A and B are invertible, Theorem 3.19(d) ensures (A ⊗ B)−1 = A−1 ⊗ B −1 .  Proof of Proposition 3.M. (a) This is immediate by the equation in Theorem 3.19(h) — union of subspaces is included in their sum — and so the inclusion is proper whenever both subspaces are nonzero. (b) This comes from item (a) and Corollary 3.10. (c) If N (λI − A) = {0} or N (μI − B) = {0}, then N (λI − A) ⊗ N (μI − B) = {0} and the inclusion is trivial. So suppose N (λI − A) = {0} and N (μI − B) = {0}. If  ∈ N (λI − A) ⊗ N (μI − B) is nonzero, then consider a representation  =  i xi ⊗ yi of it with all single tensors xi ⊗ yi nonzero. Then 0 = xi ∈ N (λI − A) and 0 = yi ∈ N (μI − B), and so λxi = Axi and μyi = Ayi , for every integer i. Then λμ(x i ⊗ yi ) = λx i ⊗ μyi = Axi ⊗ Byi = (A ⊗ B)(xi ⊗ yi ), and hence λμ = λμ i xi ⊗ yi = i λxi ⊗ μyi = i Axi ⊗ Byi = (A ⊗ B) i xi ⊗ yi =  (A ⊗ B). Therefore the nonzero  lies in N (λμI − A ⊗ B). Proposition 3.N is readily verified by using Theorem 3.19 and the definition of tensor sum in Remark 3.20(b).

Suggested Readings Brown and Pearcy [5] Defant and Floret [15] Greub [27] Halmos [30] Jarchow [42]

Lang [66] MacLane and Birkhoff [69] Roman [81] Ryan [83] Yokonuma [94]

4 Interpretations

Tensor products do exist. The next sections exhibit concrete examples of tensor products as promised in Remark 3.2(b). These represent different, but isomorphic, means of constructing tensor products out of a pair of linear spaces.

4.1 Interpretation via Quotient Space Let X and Y be nonzero linear spaces over a field F , and take the Cartesian product S = X ×Y of X and Y. Now consider the notation and terminology of Section 1.1. Thus let S / ⊂ F S be the free linear space generated by S (i.e., the linear manifold of the linear space F S of all F -valued functions on S consisting of all functions f : S → F that vanish everywhere on the complement of some finite subset of S, so that S / is itself a linear space over F ). Consider the Hamel / consisting of characteristic basis S = {e(x,y) }(x,y)∈S for the linear space S functions e(x,y) = χ{(x,y)} = χ{x} χ{y} : X ×Y → F of all singletons at each pair of vectors (x, y) in the Cartesian product S = X ×Y. Being indexed by the set S, the Hamel basis S is naturally identified with S. Thus a formal linear n combination i=1 αi (xi , yi ) of elements n(xi , yi ) in S = X ×Y is identified (by definition) with a linear combination i=1 αi e(xi ,yi ) of functions e(xi ,yi ) from / . In this sense the free linear space S / generated by S the Hamel basis S for S is identified with the so-called linear space of all formal linear combinations n α (x , y ) of elements (x , y ) of the set S = X ×Y. In particular, recall i i i=1 i i i that the pairs of vectors (x, y) in the Cartesian product S = X ×Y are in a one-to-one correspondence with the functions e(x,y) in the Hamel basis S for the function space S / . Thus let now ≈ stand for such a natural identification: n n αi (xi , yi ) ≈ αi e(xi ,yi ) and so (x, y) ≈ e(x,y) . i=1

i=1

The Cartesian product S = X × Y, being identified with the Hamel basis S for the linear space S / , is then regarded as a subset of S / , and so we write S⊂S /

for

/. S≈S⊂S

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Kubrusly, Bilinear Maps and Tensor Products in Operator Theory, Universitext, https://doi.org/10.1007/978-3-031-34093-2 4

57

58

4. Interpretations

With the identification ≈ still in force, consider the vector addition and the scalar multiplication in the linear space S / , and their respective formal addition and formal scalar multiplication of elements in S = X ×Y as follows. For every x, x1 , x2 ∈ X , every y, y1 , y2 ∈ Y, and every α ∈ F , (x1 , y1 ) + (x2 , y2 ) ≈ e(x1 ,y1 ) + e(x2 ,y2 ) = χ{(x1 ,y1 )} + χ{(x2 ,y2 )} , α (x , y) ≈ α e(x,y) = α χ{(x,y)} , where each e(x,y) = χ{(x,y)} = χ{x} χ{y} is the product of characteristic functions of singletons at x ∈ X and y ∈ Y. Then consider the differences (i)

e(x1 +x2 ,y) − e(x1 ,y) − e(x2 ,y) ≈ (x1 + x2 , y) − (x1 , y) − (x2 , y),

(ii)

e(x,y1 +y2 ) − e(x,y1 ) − e(x,y2 ) ≈ (x , y1 + y2 ) − (x , y1 ) − (x , y2 ),

(iii)

e(αx,y) − α e(x,y) ≈ (αx, y) − α(x, y),

(iv)

e(x,αy) − α e(x,y) ≈ (x, αy) − α(x, y).

It is clear that the above differences are not null. If they were, then we might identify a bilinear rule functioning for the ordered pairs (x, y) ∈ X ×Y. Roughly speaking, this is what we would like to have. Thus we will, in some fashion, identify the ordered pairs (x, y) so as to make those differences be included in some equivalence class of elements in S that is associated with some equivalence class in S / that contains the origin of S / . To begin approaching this identification scheme, we will look at the cosets [f ] of functions f in S / or, equivalently, at an equivalence class [(x, y)] of pairs (x, y) in S = X ×Y with respect to some suitable equivalence relation. A convenient equivalence relation gathering those differences at the origin of some appropriate further linear space generated by S / is naturally obtained through quotient spaces. Thus take the Cartesian product X ×Y of the linear spaces X and Y. With S / standing for the free linear space generated by S = X ×Y, consider the linear manifold M of S / generated by the differences in (i) to (iv) above,  M = span e(x1 +x2 ,y) − e(x1 ,y) − e(x2 ,y) , e(x,y1 +x2 ) − e(x,y1 ) − e(x,y2 ) ,  e(αx,y) − α e(x,y) , e(x,αy) − α e(x,y) . Since a pair (x, y) is identified with the element e(x,y) of the Hamel basis for S / (i.e., (x, y) ≈ e(x,y) ), as formal linear combinations of pairs in X ×Y are identified with linear combinations of functions in S / , we may identify the above differences of functions in S / with formal differences of pairs in S:  M ≈ span (x1 + x2 , y) − (x1 , y) − (x2 , y) , (αx, y) − α (x, y) ,  (x, y1 + y2 ) − (x, y1 ) − (x, y2 ) , (x, αy) − α(x, y) . Now consider the quotient space S / /M of S / modulo M. It will be shown in Theorem 4.1 that the quotient space S / /M is a tensor product space (sometimes called the algebraic tensor product) of X and Y associated with a natural

4.1 Interpretation via Quotient Space

59

bilinear map θ (which will be defined shortly). Take an arbitrary pair (x, y) in the Cartesian product X ×Y, let e(x,y) = χ{(x,y)} be the corresponding characteristic function in the linear space S / , and consider the natural quotient map (which is surjective) π: S /→S / /M such that π(f ) = [f ] = f + M in S / /M for every f in S / . Define the map θ : X ×Y → S / /M as follows. For every (x, y) ∈ X ×Y, set θ(x, y) = π(e(x,y) ). Therefore θ(S) = R(θ) = R(π|S ) = π(S ). Since the Hamel basis S = {e(x,y) }(x,y)∈S for the free linear space S / generated by the set S is naturally identified with S itself, S = {e(x,y) }(x,y)∈S ≈ S = X ×Y, then the domain S of θ is identified with the domain S of π|S . Also, since their values coincide, θ(x, y) = π(e(x,y) ), then θ is naturally identified with the restriction π|S of the natural quotient map π to the Hamel basis S . So write θ = π|S

for

θ ≈ π| S .

Denote by x ⊗ y the image of e(x,y) ∈ S ⊂ S / ≈ (x, y) ∈ X ×Y = S under π, x ⊗ y = π(e(x,y) ) = [e(x,y) ] = e(x,y) + M ≈ [(x, y)], where [(x, y)] is the equivalence class of all pairs (x, y) for which e(x,y) belongs to the equivalence class [e(x,y) ]. So x ⊗ y is identified with [(x, y)]. Each pair (x, y) in X ×Y regarded as an element of S / (i.e., identified with an element of the Hamel basis S = {e(x,y) }(x,y)∈X ×Y for the linear space S / ) is projected (through the quotient map π : S /→S / /M) onto an element of S / /M denoted by x ⊗ y. Elements of S / /M of the form x ⊗ y are the images of θ : X ×Y → S / /M, and are again referred to as single tensors or decomposables: x ⊗ y = θ(x, y)

for every

(x, y) ∈ X ×Y.

With S / standing for the free linear space generated by the Cartesian product X ×Y of linear spaces X and Y, and M standing for the linear manifold generated by the differences in (i) to (iv), we show that S / /M is in fact a tensor product space of X and Y associated with the bilinear map θ : X ×Y → S / /M. Theorem 4.1.

(S / /M, θ) is a tensor product of X and Y .

Proof. Consider the axioms (a) and (b) in Definition 3.1, where X × Y is the Cartesian product of linear spaces X and Y. (a1 ) By the definition of M, the differences in (i) to (iv) lie in M = [0] ∈ S / /M. Thus for every x, x1 , x2 ∈ X , every y, y1 , y2 ∈ Y, and every α ∈ F ,

60

4. Interpretations

θ(x1 + x2 , y) = [e(x1 +x2 ,y) ] = [e(x1 ,y) ] + [e(x2 ,y) ] = θ(x1 , y) + θ(x2 , y), θ(x, y1 + y2 ) = [e(x,y1 +y2 ) ] = [e(x,y1 ) ] + [e(x,y2 ) ] = θ(x, y1 ) + θ(x, y2 ), θ(αx, y) = [e(αx,y) ] = α[e(x,y) ] = αθ(x, y), θ(x, αy) = [e(x,αy) ] = α[e(x,y) ] = αθ(x, y). Hence θ : X ×Y → S / /M is a bilinear map (and so αx⊗y = α(x⊗y) = x⊗αy). (a2 ) Since S = {e(x,y) }(x,y)∈X ×Y is a Hamel basis for the linear space S / , then / /M (Remark 1.5). Thus, as the ranges coincide, R(θ) = R(π|S ), span π(S ) = S / /M. span R(θ) = span R(π|S ) = span π(S ) = S (b) Take a bilinear map φ : X ×Y → Z into a linear space Z and consider the  on the free linear space S transformation Φ / generated by S = X ×Y, : S Φ / → Z, associating a bona fide linear / n combination of vectors from Z with each f ∈ S as follows. For each f = i=1 αi e(xi ,yi ) ∈ S / , set  )= Φ(f

n i=1

αi φ(xi , yi ) ∈ Z,

 between the linear spaces S which defines a linear transformation Φ / and Z. n m (Indeed, for every f = i=1 αi (xi , yi ) and g = i=1 βi (ui , vi ) in S / we get n m     Φ(f + g) = i=1 αi φ(xi , yi ) + i=1 βi φ(xi , yi ) = Φ(f ) + Φ(g) and Φ(αf )= n  αα φ(x , y ) = α Φ(f ) — regardless of whether φ is bilinear or not.) i i i i=1  (x ,y ) ) = φ(x, y), then Moreover, since Φ(e i i  S ) = φ(S). Φ(  S is naturally identified with φ. So write Since S ≈ S = X ×Y, then Φ|  S = φ. Φ| Furthermore, since φ : X ×Y → Z is a bilinear map, the linear transformation : S Φ / → Z evaluated at the differences in (i) to (iv) is null, and hence by def  inition of the linear manifold M of S / we get Φ(M) = 0. That is, M ⊆ N (Φ). Then according to Theorem 1.4(a) there is a unique linear transformation Φ: S / /M → Z  = Φ ◦ π. Now restricting to S ⊂ S such that Φ / , and since span π(S ) = S / /M,  S = (Φ ◦ π)| S = Φ| spanR(π) ◦ π|S = Φ ◦ θ. Equivalently, the diagrams we get φ = Φ|

4.1 Interpretation via Quotient Space  Φ

S / −−−→ Z  ⏐  ⏐Φ π

61

φ

S −−−→ Z 

and

θ



 ⏐ ⏐Φ

S / /M

S / /M

commute. If we again identify S ≈ S (thus regarding S = X ×Y as a subset of  S for φ = Φ|  S), then the diagram S / and writing θ = π|S for θ = π| S and φ = Φ| φ

X ×Y −−−→ Z  ⏐  ⏐Φ θ S / /M commutes, and so the pair (S / /M, θ) satisfies the axioms of Definition 3.1.



Thus consider the tensor product space X ⊗ Y = S / /M of two linear spaces X and Y. Let  denote an arbitrary element in S / /M so that  = [f ] ∈ S / /M, where f ∈ S / is any representative of the equivalence class [f ]. Since S = / , then take the expansion f = {e(x,y) }(x,y)∈X ×Y is a Hamel basis for S  n α e of f in terms of {e } (x,y) (x,y)∈X ×Y to get i=1 i (xi ,yi )

n n n  = [f ] = αi e(xi ,yi ) = αi [e(xi ,yi ) ] = αi (xi ⊗ yi ). i=1

i=1

i=1

Recall from Remark 1.5 that a Hamel basis for S / /M is in a one-to-one correspondence with a Hamel basis for any algebraic complement of M in S /, and therefore a Hamel basis for S / /M is included in a Hamel basis for S / . As we saw above, the set {x ⊗ y}(x,y)∈X ×Y = {[e(x,y) ]}(x,y)∈X ×Y , which is in a /, one-to-one correspondence with the Hamel basis S = {e(x,y) }(x,y)∈X ×Y for S spans S / /M. So it includes a Hamel basis for S / /M. Thus there exists a Hamel basis for the tensor product space S / /M consisting of single tensors only, say {xγ ⊗ yγ }γ∈Γ ⊆ {x ⊗ y}(x,y)∈X ×Y , in accordance with Theorem 3.9. In fact, since αi (xi ⊗ yi ) = αi xi ⊗ yi = xi ⊗ αi yi , write the above expansion as    = [f ] = x⊗y = xγ ⊗ yγ , both sums being finite, the first one drawn from the collection of all single tensors {x ⊗ y}(x,y)∈X ×Y and the second from a Hamel basis {xγ ⊗ yγ }γ∈Γ for S / /M, as in Corollary 3.4. Also, dim S / /M = dim X · dim Y by Corollary 3.10. This interpretation where single tensors are defined as equivalence classes of characteristic functions of singletons in the Cartesian product of linear spaces is referred to as the quotient space approach. Remark 4.2. A Summary of the Quotient Space Approach. S = X ×Y: the Cartesian product of linear spaces X and Y.

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4. Interpretations

S / ⊂ F S : the linear space of functions vanishing outside finite subsets of S. e(x,y) = χ{(x,y)} : F X ×Y → F . / : a Hamel basis for S /. S = {e(x,y) }(x,y)∈X ×Y ⊂ S /. S≈S⊂S /; M: the linear manifold of S / spanned by specific differences of e(x,y) ∈ S those differences that prevent the rule (x, y) → e(x,y) to be bilinear. / /M S / /M: the quotient space where the equivalence classes [e(x,y) ] ∈ S are such that the rule (x, y) → [e(x,y) ] is bilinear. X ⊗Y =S / /M. π(f ) = [f ]: the natural quotient map π : S /→S / /M. / /M. θ = π|S : the natural bilinear map θ : X ×Y → S / /M (a single tensor). x ⊗ y = θ(x, y) = π(e(x,y) ) = [e(x,y) ] ∈ S  / /M ⇐⇒  = [f ] = x ⊗ y (a finite sum).  = [f ] ∈ S Such a realization of the tensor product via quotient space has naturally been extended from linear spaces over fields to modules over rings (see, e.g., [66, Chapter XVI]), where a tensor product is regarded as a universal object in the realm of category theory.

4.2 Interpretation via Linear Maps of Bilinear Maps Let X and Y be nonzero linear spaces over the same field F . Given an arbitrary but fixed linear space F over F , consider the linear space b[X ×Y, F] of all bilinear maps ψ : X ×Y → F of the Cartesian product X ×Y into F (in particular, ψ may be acting on the direct sum X ⊕ Y). Associated with each pair (x, y) ∈ X ×Y, consider a transformation x ⊗ y : b[X ×Y, F] → F defined by (x ⊗ y)(ψ) = ψ(x, y) ∈ F

for every

ψ ∈ b[X ×Y, F].

This again is referred to as a single tensor , which is a linear transformation. (In fact, since b[X ×Y, F] is a linear space, the sum of bilinear maps is a bilinear map, and so it makes sense to evaluate (x ⊗ y)(ψ1 + ψ2 ) = (ψ1 + ψ2 )(x, y) = ψ1 (x, y) + ψ2 (x, y) = (x ⊗ y)(ψ1 ) + (x ⊗ y)(ψ2 ) for every ψ1 , ψ2 ∈ b[X ×Y, F]. Also, α(x ⊗ y)(ψ) = (x ⊗ y)(αψ) for every ψ ∈ b[X ×Y, F] and α ∈ F .) Thus x ⊗ y ∈ L[ b[X ×Y, F], F], a linear map of bilinear maps, where L[ b[X ×Y, F], F] is the linear space of all linear transformations from b[X ×Y, F] to F. Hence a single tensor x ⊗ y associated with a pair (x, y) is defined as a linear transformation of bilinear maps given by evaluation at the pair

4.2 Interpretation via Linear Maps of Bilinear Maps

63

(i.e., (x ⊗ y)(ψ) = ψ(x, y)), and therefore this approach to tensor product focuses explicitly and precisely on the linearization of bilinear maps. Now take the collection X ,Y, F of all single tensors as defined above,   X ,Y, F = x ⊗ y ∈ L[ b[X ×Y, F], F] : x ∈ X and y ∈ Y , consider its span, which is a linear manifold of the linear space L[ b[X×Y, F], F], spanX ,Y, F ⊆ L[ b[X ×Y, F], F], and define the map θ : X ×Y → X ,Y, F ⊆ spanX ,Y, F ⊆ L[ b[X ×Y, F], F] as follows: for each pair (x, y) ∈ X ×Y set θ(x, y) = x ⊗ y. Thus θ(x, y) ∈ L[b[X ×Y, F], F] is such that θ(x, y)(ψ) = ψ(x, y) ∈ F for every ψ ∈ b[X ×Y, F]. The linear space spanned by the single tensors, spanX ,Y, F , is a tensor product space of X and Y associated with the natural bilinear map θ : X ×Y → R(θ) = X ,Y, F ⊆ spanX ,Y, F = span R(θ). This is shown below. Theorem 4.3.

(spanX ,Y, F , θ) is a tensor product of X and Y .

Proof. Take an arbitrary linear space F. Consider the axioms (a) and (b) in Definition 3.1 where X ×Y is the Cartesian product of linear spaces X and Y. (a1 ) The map θ : X×Y → spanX ,Y, F is bilinear (i.e., θ ∈ b[X×Y, spanX ,Y, F ] — this is the bilinear map θ : X ×Y → L[ b[X ×Y, F], F] in Proposition 2.L.) Indeed, take x, x1 , x2 ∈ X , y, y1 , y2 ∈ Y, and α ∈ F arbitrarily. Also take an arbitrary bilinear map ψ ∈ b[X ×Y, F]. By the definition of the single tensor x ⊗ y, and since ψ is bilinear, we get ((x1 + x2 ) ⊗ y)(ψ) = ψ(x1 + x2 , y) = ψ(x1 , y) + ψ(x2 , y) = (x1 ⊗ y)(ψ) + (x2 ⊗ y)(ψ), Therefore (x1 + x2 ) ⊗ y = (x1 ⊗ y) + (x2 ⊗ y). Similarly, x ⊗ (y1 + y2 ) = (x ⊗ y1 ) + (x ⊗ y2 ). Moreover, α(x ⊗ y)(ψ) = αψ(x, y) = ψ(αx, y) = ψ(x, αy) = (αx ⊗ y)(ψ) = (x ⊗ αy)(ψ), and hence α(x ⊗ y) = αx ⊗ y = x ⊗ αy. Thus by the definition of θ, θ(x1 + x2 , y) = θ(x1 , y) + θ(x2 , y),

θ(x, y1 + y2 ) = θ(x, y1 ) + θ(x, y2 ),

α θ(x, y) = θ(α x, y) = θ(x, α y). (a2 ) Since θ(x, y) = x ⊗ y, then R(θ) = X ,Y, F . So span R(θ) = spanX ,Y, F . (b) An arbitrary element  of the linear space spanX ,Y, F is a (finite) linear combination of single tensors, and therefore it is a linear transformation in L[ b[X ×Y, F], F]. Since α(x ⊗ y) = αx ⊗ y = x ⊗ αy for every α ∈ F , every  in spanX ,Y, F is a finite sum of single tensors:

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4. Interpretations

=

 i

xi ⊗ yi ∈ spanX ,Y, F ⊆ L[ b[X ×Y, F], F].

Given an arbitrary bilinear map φ ∈ b[X ×Y, Z] into any linear space Z over

F , consider the transformation

Φ : spanX ,Y, F → Z associating a sum of vectors in the linear space Z with each sum of single tensors in the linear space spanX ,Y, F as follows.   φ(xi , yi ) ∈ Z for every  = xi ⊗ yi ∈ spanX ,Y, F . Φ() = i

i

(Note: φ(xi , yi ) lies in Z for a given bilinear map φ of X ×Y into any Z, while (xi ⊗ yi )(ψ) = ψ(xi , yi ) lies in F for every bilinear map ψ of X ×Y into a fixed F.) As is readily verified, Φ is a linear transformation: Φ ∈ L[ spanX ,Y, F , Z].       (Indeed, for  = k and α ∈ F we get Φ( +  ) =  i xi ⊗yi , = k xk ⊗ y     + k xk ⊗ yk ) =  ) + Φ( ) and Φ( i xi ⊗ yi i φ(xi , yi ) + k φ(xk , yk ) = Φ(  Φ(α ) = Φ( i α(xi ⊗ yi )) = Φ( i αxi ⊗ yi ) = i φ(αxi , yi ) = α i φ(xi , yi ) = αΦ() — since φ is a bilinear map.) Moreover, for every (x, y) ∈ X ×Y,  (Φ ◦ θ)(x, y) = Φ θ(x, y) = Φ(x ⊗ y) = φ(x, y) and so φ = Φ ◦ θ, leading to the commutative diagram φ

X ×Y

−−−→ θ





Z  ⏐ ⏐Φ

spanX ,Y, F , and the pair (spanX ,Y, F , θ) satisfies the axioms of Definition 3.1.



This interpretation where single tensors are defined as linear transformations of bilinear maps is referred to as the linear-bilinear approach, and it is finely tailored to highlight the central property of tensor products as a tool to linearize bilinear maps according to Theorem 3.7. This is the reason why the proof of Theorem 4.3 is simpler than the proof of its quotient space counterpart in Theorem 4.1. Remark 4.4. A Summary of the Linear-Bilinear Approach. b[X ×Y, F] = {ψ ∈ F X ×Y : ψ is a bilinear map}: a linear space. (x ⊗ y)(ψ) = ψ(x, y): a single tensor x ⊗ y : b[X ×Y, F] → F.

4.3 Variants of the Linear-Bilinear Approach

65

x ⊗ y ∈ L[ b[X ×Y, F], F] (x ⊗ y is a linear transformation). X ,Y, F = {x ⊗ y ∈ L[ b[X ×Y, F], F] : x ∈ X and y ∈ Y}. X ⊗ Y = spanX ,Y, F ⊆ L[ b[X ×Y, F], F]. θ ∈ b[X ×Y, X ⊗ Y]: the natural bilinear map θ : X ×Y → X ⊗ Y. θ(x, y) = x ⊗ y ∈ X ,Y, F ⊆ spanX ,Y, F = X ⊗ Y ⊆ L[ b[X ×Y, F], F].   ∈ spanX ,Y, F ⇐⇒  = x ⊗ y (a finite sum), and so an element  of the tensor product space X ⊗ Y = spanX ,Y, F lies in the linear space L[ b[X ×Y, F],F] of all linear transformations of bilinear maps,  ∈ X ⊗ Y = spanX ,Y, F ⊆ L[ b[X ×Y, F], F].

4.3 Variants of the Linear-Bilinear Approach An important particular case of Theorem 4.3 refers to linear and bilinear forms by setting F = F so that single tensors are linear forms of bilinear forms, x ⊗ y ∈ X ,Y, F ⊆ L[b[X ×Y, F ], F ] = b[X ×Y, F ] , with b[X ×Y, F ] standing for the dual of the linear space b[X ×Y, F ]. Now consider a slightly different procedure, still referring to the particular case of F = F , where bilinear forms ψ are restricted to a subset of b[X ×Y, F ] consisting of bilinear forms which are products of linear forms f and g, ψ(x, y) = f (x) g(y)

for some

f ∈ X = L[X , F ] and g ∈ Y = L[Y, F ].

Such a subset of the linear space b[X ×Y, F ], namely,   bX  ×Y  [X ×Y, F ] = ψ ∈ b[X ×Y, F ] : ψ(x, y) = f (x) g(y) for (f, g) ∈ X ×Y , is not a linear manifold of b[X ×Y, F ] (indeed, a sum of two bilinear forms in bX  ×Y  [X ×Y, F ] may not lie in bX  ×Y  [X ×Y, F ]). Define single tensors as before: to each pair (x, y) in the Cartesian product X×Y associate a function x ⊗ y : bX  ×Y  [X ×Y, F ] → F defined for every ψ ∈ bX  ×Y  [X ×Y, F ] by (x⊗y)(f, g) = (x⊗y)(ψ) = ψ(x, y) = f (x) g(y)

for every

(f, g) ∈ X ×Y .

The difference between this and the previous procedure is due to the fact that single tensors are not linear transformations (or linear forms) any longer since their domain bX  ×Y  [X ×Y, F ] is not a linear manifold of the linear space b[X ×Y, F ], thus not a linear space. However, they can now be regarded as bilinear forms on the Cartesian product of the linear spaces X and Y . (In fact, we get (x ⊗ y)(f1 + f2 , g) = (f1 + f2 )(x) g(y) = (f1 (x) + f2 (x)) g(y) = f1 (x) g(y) + f2 (x) g(y) = (x ⊗ y)(f1 , g) + (x ⊗ y)(f2 , g) and, on the other hand,

66

4. Interpretations

(x ⊗ y)(f, g1 + g2 ) = (x ⊗ y)(f, g1 ) + (x ⊗ y)(f, g2 ); moreover (x ⊗ y)(αf, g) = α(x ⊗ y)(f, g) = (x ⊗ y)(f, αg), for every f1 , f2 , f ∈ X , every g1 , g2 , g ∈ Y , and every α ∈ F ; for each pair (x, y) ∈ X ×Y.) Hence

x ⊗ y ∈ b[X ×Y , F ] = b[L[X , F ]×L[Y, F ], F ].

 Thus take the collection X,Y of all these single tensors    X,Y = x ⊗ y ∈ b[X ×Y , F ] : x ∈ X and y ∈ Y ,  and consider its span, spanX,Y , which is now a linear manifold of the linear space b[X ×Y , F ] of all bilinear forms of pairs of linear forms,  spanX,Y ⊆ b[X ×Y , F ].  As before, define the map θ  : X ×Y → X,Y for each pair (x, y) ∈ X ×Y by

θ (x, y) = x ⊗ y. Then the value of θ  at (x, y) ∈ X ×Y is a bilinear form, θ (x, y) ∈ b[X ×Y , F ], which is given by θ (x, y)(f, g) = f (x) g(y) ∈ F for every (f, g) ∈ X ×Y and,   ⊆ spanX,Y ⊆ b[X ×Y , F ] is a bilinear map. as before, θ  : X ×Y → X,Y Corollary 4.5.

 , θ ) is a tensor product of X and Y. (spanX,Y

Proof. Replace F by F in the proof of Theorem 4.3 so that L[ b[X ×Y, F], F] is replaced by L[ b[X ×Y, F ], F ]. Then replace the linear space b[X ×Y, F ] by the subset bX  ×Y  [X ×Y, F ], still keeping the same definition of single tensors, and replace X ,Y,F ⊆ L[ b[X ×Y, F], F] by X ,Y ⊆ b[L[X , F ]×L[Y, F ], F ]. Again, θ  : X ×Y → spanX ,Y is a bilinear map with span R(θ ) = spanX ,Y — this is the bilinear map θ  : X ×Y → b[X ×Y , F ] in Proposition 2.M. These modifications do not alter the argument in the proof of Theorem 4.3, which still works when to each bilinear map φ : X ×Y → Z there is associated the  . 

same linear transformation Φ into Z now acting on spanX,Y Remark 4.6. Single Tensors in Hilbert Space. A common and useful example of such a particular case goes as follows. Let X and Y be Hilbert spaces (either both real or both complex) with inner products · ; ·X and · ; ·Y , which are sesquilinear forms (but not bilinear forms in the complex case). Here algebraic duals X and Y are naturally replaced by topological duals X ∗ and Y ∗ of continuous linear functionals. (Hilbert spaces are discussed in Section 5.4 and the present example is revisited at the beginning of Section 9.4). A single tensor x ⊗ y associated with a pair (x, y) in X ×Y is usually defined in this case as (x ⊗ y)(u, v) = x ; uX y ; vY

for every

(u, v) ∈ X ×Y.

The Riesz Representation Theorem for Hilbert spaces (stated in Proposition 5.U) ensures that f lies in X ∗ and g lies in Y ∗ if and only if f (·) = · ; uX and

4.3 Variants of the Linear-Bilinear Approach

67

g(·) = · ; vY for some u in X and v in Y. Thus identify f ∈ X ∗ and g ∈ Y ∗ with u ∈ X and v ∈ Y such that the pair (u, v) ∈ X ×Y is identified with the pair (f, g) ∈ X ∗ ×Y ∗. Then a single tensor x ⊗ y associated with a pair (x, y) in X ×Y is a bilinear form x ⊗ y : X ∗×Y ∗ → F on X ∗×Y ∗, not on X ×Y — i.e., x ⊗ y ∈ b[X ∗ ×Y ∗, F ], which is equivalently written (as in Corollary 4.5) by (x ⊗ y)(f, g) = f (x) g(y)

for every

(f, g) ∈ X ∗ ×Y ∗ .

By Theorem 4.3 with F = F we get spanX ,Y, F ⊆ b[X ×Y, F ] . Moreover,  ⊆ according to Corollary 4.5 (which assumes F = F a priori ) we got spanX,Y b[X ×Y , F ]. So there are realizations of tensor product spaces of linear spaces X and Y which can be included in the algebraic dual of the linear space of all bilinear forms on the Cartesian product X ×Y, or in the linear space of all bilinear forms on the Cartesian product X ×Y of their algebraic duals. We consider further possibilities as follows. For each (x, y) ∈ X ×Y take a transformation x ⊗ y : X → Y defined by (x ⊗ y)(f ) = f (x)y ∈ Y

for every

f ∈ X .

This is a linear transformation of the linear space X (i.e., of the algebraic dual of the linear space X ) onto the linear space Y. (In fact, (x ⊗ y)(f1 + f2 ) = (f1 + f2 )(x)y = (f1 (x) + f2 (x))y = f1 (x)y + f2 (x)y = (x ⊗ y)(f1 ) + (x ⊗ y)(f2 ) and (x ⊗ y)(αf ) = αf (x)y = α(x ⊗ y)(f ) for every f, f1 , f2 ∈ X and every α ∈ F — for each pair (X , Y) ∈ X ×Y.) Therefore 

(x ⊗ y) ∈ L[X , Y].

  = x ⊗ y ∈ L[X , Y] : x ∈ X and y ∈ Y , take its span As before, set X,Y  ⊆ L[X , Y], spanX,Y  as follows: for every (x, y) ∈ X ×Y, and define the map θ: X ×Y → spanX,Y

θ(x, y) = x ⊗ y. Now the value of θ at (x, y) ∈ X ×Y is a linear transformation, θ(x, y) in L[X , Y], which is given by θ(x, y)(f ) = f (x)y ∈ Y for every f ∈ X . Corollary 4.7.

 , θ ) is a tensor product of X and Y. (spanX,Y

Proof. Consider the argument in the proof of Theorem 4.3. (a) Since f is a linear functional and Y is a linear space, θ: X ×Y → spanX ,Y ⊆ L[X , Y]  is a bilinear map whose range satisfies the identity span R(θ ) = spanX,Y (the bilinear map in Proposition 2.N). Also, (b) for an arbitrary bilinear map φ : X ×Y → Z into any linear space Z over F , take the same linear transfor mation Φ : spanX,Y → Z defined in the proof of Theorem 4.3, now acting on   spanX,Y . Again, φ = Φ ◦ θ and the axioms of Definition 3.1 are fulfilled.

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4. Interpretations

Symmetrically, for each (x, y) ∈ X ×Y define x ⊗ y : Y → X by (x ⊗ y)(g) = g(y)x ∈ X

for every

g ∈ Y .

Again this is a linear transformation, now of Y onto X , and so (x ⊗ y) ∈ L[Y , X ].    Thus set X,Y = x ⊗ y ∈ L[Y , X ] : x ∈ X and y ∈ Y , take its span  spanX,Y ⊆ L[Y , X ],  and define the map θ : X ×Y → spanX,Y for every (x, y) ∈ X ×Y by

θ(x, y) = x ⊗ y. This time the value of θ at (x, y) ∈ X ×Y is the linear transformation θ(x, y) in L[Y , X ] given by θ(x, y)(g) = g(y)x ∈ X for every g ∈ Y . Corollary 4.8.

 , θ ) is a tensor product of X and Y. (spanX,Y

Proof. A symmetrical argument as in the proof of Corollary 4.7 (the bilinear 

map θ : X ×Y → L[Y , X ] is the one in Proposition 2.O). Let X and Y be linear spaces, let T be a tensor product space of X and Y, and let L be another linear space, all over the same field. Since tensor product spaces for the same pair of linear spaces are isomorphic (Corollary 3.6), they are unique up to an isomorphism, and we have referred to the tensor product space of X and Y by denoting it by X ⊗ Y. This means that X ⊗ Y ∼ = T for every tensor product space T for X and Y. Thus if T ⊆ L, as we had in Theorem 4.3 and Corollaries 4.5 and 4.7, then X ⊗Y ∼ = T ⊆ L, and so every tensor product space of X and Y is embedded in L, and we write X ⊗Y ⊆L

for

X ⊗ Y → L.

Remark 4.9. Embeddings of Tensor Products. Some inclusions (embeddings) of tensor product spaces into well-known linear spaces are summarized below. (a) X ⊗ Y ⊆ b[X ×Y, F ] (with (x ⊗ y)(ψ) = ψ(x, y) in Theorem 4.3 for F = F ). (b) X ⊗ Y ⊆ b[X ×Y , F ] (with (x ⊗ y)(f, g) = f (x) g(y) in Corollary 4.5). (c) X ⊗ Y ⊆ L[X , Y] (with (x ⊗ y)(f ) = f (x)y in Corollary 4.7).

4.3 Variants of the Linear-Bilinear Approach

69

(d) X ⊗ Y ⊆ L[Y , X ] (with (x ⊗ y)(g) = g(y)x in Corollary 4.8). (e) X ⊗ Y ⊆ (X ⊗ Y) (Remark 3.18). (f) X ⊗ Y ⊆ b[X ×Y, F ] (by item (e) since (X ⊗ Y) ∼ = b[X ×Y, F ] according to Remark 3.8(b)). (g) X ⊗ Y ⊆ L[X , Y ] (set (f ⊗ g)(x) = f (x)g, argument of Corollary 4.7; or by item (f) since b[X ×Y, F ] ∼ = L[X , Y ] according to Remark 2.7(b)). (h) X ⊗ Y ⊆ L[Y, X ] (set (f ⊗ g)(y) = g(y)f , argument of Corollary 4.8); or by item (g) since L[X , Y ] ∼ = L[Y, X ] according to Remark 2.7(b)). (i) X ⊗ Y ⊆ L[X , Y] (set (f ⊗ y)(x) = f (x)y, argument of Corollary 4.7). (j) X ⊗ Y ⊆ L[Y , X ] (replace X by X in item (d)). (k) X ⊗ Y ⊆ b[X ×Y , F ] (by item (j) since L[Y , X ] ∼ = b[X ×Y , F ] according to Remark 2.7(b)). () X ⊗ Y ⊆ L[Y, X ] (set (x ⊗ g)(y) = g(y)x, argument of Corollary 4.8). (m) X ⊗ Y ⊆ L[X , Y ] (replace Y by Y in item (c)). (n) X ⊗ Y ⊆ b[X ×Y, F ] (by item (m) since L[X , Y ] ∼ = b[X ×Y, F ] according to Remark 2.7(b)).  By Corollary 4.5 the tensor product space spanX,Y of X and Y is included in the linear space b[X ×Y , F ],  spanX,Y ⊆ b[X ×Y , F ].

Thus according to Corollary 3.6 every tensor product space of X and Y is isomorphic to spanX ,Y , and therefore is embedded in b[X ×Y , F ] (cf. Remark 4.9(b)). In particular, the tensor product space of Theorem 4.3, spanX ,Y, F ⊆ L[ b[X ×Y, F], F], (and so spanX ,Y, F ⊆ b[X ×Y, F ] ) is embedded in b[X ×Y , F ]: spanX ,Y, F → b[X ×Y , F ]

70

4. Interpretations

for every linear space F. And all this comes from Theorem 3.5 which, through Corollary 3.6, established the embeddings summarized in Remark 4.9. There is, however, a special configuration of Theorem 4.3 (by considering a particular concrete realization for the linear space Z) which is able to supply an alternate way of exhibiting the above embedding without using the isomorphism between all tensor product spaces of X and Y established in Theorem 3.5. The construction of such an alternate procedure is not as neat as the isomorphism procedure of Remark 4.9 that uses Theorem 3.5, but it has an interest by itself due to the technique applied in its proof. This is shown below. Corollary 4.10. Let X , Y, F be nonzero linear spaces over the same field F . Consider the tensor product (spanX ,Y, F , θ) of X and Y as in Theorem 4.3. There exist a particular linear space Z, viz., Z = b[X ×Y , F ], and a bilinear map φ in b[X ×Y, Z] whose associated linear transformation Φ in L[spanX ,Y, F , Z] is injective. Thus Φ embeds spanX ,Y, F into Z: Φ : spanX ,Y, F → b[X ×Y , F ]. Proof. Fix a pair (x, y)∈X×Y. Take arbitrary linear forms f ∈X and g ∈ Y . Consider the functional φ(x, y) : X ×Y → F given for every(f, g) ∈ X ×Y by φ(x, y)(f, g) = f (x) g(y) ∈ F . As we saw before (related to Corollary 4.5), this is a bilinear form: φ(x, y) ∈ b[X ×Y , F ]. Since b[X ×Y , F ] is a linear space, set Z = b[X ×Y , F ] and consider the map φ : X ×Y → Z = b[X ×Y , F ] with φ(x, y) ∈ b[X ×Y , F ] defined above. As in Corollary 4.5, φ is bilinear: φ ∈ b[X ×Y, Z] = b[X ×Y, b[X ×Y , F ]]. Take an arbitrary linear space F over F and consider the tensor product (spanX ,Y, F , θ) of linear spaces X and Y as in Theorem 4.3. Thus by Definition 3.1 and Corollary 3.4 there is a unique linear transformation Φ ∈ L[spanX ,Y, F , Z] = L[spanX ,Y, F , b[X ×Y , F ]]

4.3 Variants of the Linear-Bilinear Approach

71

for which the diagram X ×Y

φ

−−−→ 

θ



Z = b[X ×Y , F ]  ⏐ ⏐Φ

spanX ,Y, F commutes, where the tensor product space spanX ,Y, F of X and Y is included in L[ b[X ×Y, F], F] and the natural bilinear map θ in b[X ×Y, spanX ,Y, F ] is given by θ(x, y) = x ⊗ y ∈ L[b[X ×Y, F], F] for each (x, y) ∈ X ×Y, with the single tensor x ⊗ y defined by (x ⊗ y)(ψ) = ψ(x, y) ∈ F for ψ ∈ b[X ×Y, F]. Hence θ(x, y)(ψ) = ψ(x, y) ∈ F for every ψ ∈ b[X ×Y, F]. Moreover, the linear transformation Φ is such that for every  = i xi ⊗ yi ∈ spanX ,Y, F  Φ() = φ(xi , yi ) ∈ Z = b[X ×Y , F ] i

(cf. proof of Theorem 4.3), and so for every (f, g) ∈ X ×Y   Φ()(f, g) = φ(xi , yi )(f, g) = f (xi ) g(yi ) ∈ F . i i n Observe that  = 0 ⇐⇒  = i=1 xi ⊗ yi = 0 with xi = 0 and yi = 0 for all i = 1, . . . , n for some n ≥ 1 (since x ⊗ y = 0 if either x = 0 or y = 0 because n αx ⊗ y = α(x ⊗ y) = x ⊗ αy). Also observe that for  = i=1 xi ⊗ yi n Φ() = 0 ⇐⇒ Φ()(f, g) = 0 ∀f, g ⇐⇒ f (xi ) g(yi ) = 0 ∀f, g. i=1

n

⊗ yi = 0 =⇒ for some f, g. i=1 f (xi ) g(yi ) = 0 n Proof of Claim. Suppose i=1 xi ⊗ yi = 0. Since θ(x, y) = x ⊗ y and θ is bilinear, we may assume the finite sets {xi }ni=1 and {yi }ni=1 are linearly independent (cf. Proposition 3.A in Section 3.4), so all single tensors xi ⊗ yi in the above sum are nonzero and n ≤ min{dim X , dim Y} is the least integer yielding the corresponding nonzero finite sum of single tensors. Recall: for and g ∈ Y for which each 0 = x ∈ X and each 0 = y ∈ Y there exist f ∈ X  n f (x) = 0 and g(y) = 0 (cf. Proposition 1.G). Also, if i=1 g(yi )xi = 0, then n gi (y) = 0 for every i = 1, . . . , n, as {xi }i=1 is linearly independent. Thus by the above italicized result there exists g such that g(yi ) = 0 for some yi in {yj }nj=1 (as {yi }ni=1 is linearly independent and so has only nonzero vectors). n Hence  n i=1 g(yi )xi = 0. So by the same n italicized result there is an f for which =  0, which means g(y )x 

f i i i=1 i=1 f (xi )g(yi ) = 0 since f is linear. Claim.

i=1 xi

n

Thus  = 0 =⇒ Φ() = 0. Then Φ is injective. So the linear transformation Φ : spanX ,Y, F → Z = b[X ×Y , F ] embeds the linear space spanX ,Y, F ⊆ L[b[X ×Y, F], F] into the linear space 

Z = b[X ×Y , F ] (cf. Remark 3.2(c)).

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4. Interpretations

Remark 4.11. Multiple Tensor Products. It is readily verified that the preceding arguments (in Chapters 3 and 4) can be naturally extended to build up the concept of an algebraic tensor product of a finite collection {Xi }ni=1 of n linear spaces over the same field, yielding a tensor product space i=1 Xi of a finite number of linear spaces. n The construction is based on the notions of multiple Cartesian products i=1 Xi , n-tuples, and multilinear maps, as natural extensions of Cartesian products of two linear spaces, ordered pairs, and bilinear maps. Actually, the results in Chapters 3 and 4 remain true (essentially with the same statement, following similar arguments) if extended to such multiple tensor product cases. It may be a significant endeavor to extend a result concerning tensor products from a pair of linear spaces to an n-tuple (or to an ∞-tuple). Sometimes this is a simple task (achieved by induction, when necessary), but not always. On the other hand, what may be not always simple is working the other way around: when a notion is primarily defined for an n-tuple, it may be wise to reduce the multiplicity down to a pair to see clearly what is really happening.

4.4 Additional Propositions Let X and Y be linear spaces over the same field F . Take their Cartesian product X ×Y. Equip X ×Y with the direct sum ⊕ (which simply means coordinatewise addition) and with standard coordinatewise scalar multiplication as usual: for every (x1 , y1 ), (x2 , y2 ), (x, y) in X × Y and every α in F , (x1 , y1 ) ⊕ (x2 , y2 ) = (x1 + x2 , y1 + y2 )

and

α(x, y) = (αx, αy).

This leads to the notion of the linear space direct sum X ⊕ Y = (X ×Y, ⊕,  , F ) whose underlying space is the Cartesian product X ×Y of linear spaces. As seen in Section 2.1, a bilinear map can be equally defined on X ×Y or on X ⊕ Y, as the algebra acquired by X ×Y when equipping it with the direct sum plays no role when dealing with bilinear maps: b[X ⊕ Y, Z] = b[X ×Y, Z] for any linear space Z. Now since a single tensor x ⊗ y is precisely the image θ(x, y) of a bilinear map θ defined on X ×Y (or on X ⊕ Y — in this context it makes no difference if an algebra equips the Cartesian product of X and Y) into a linear space T, say x ⊗ y = θ(x, y) for every (x, y) ∈ X ×Y, we get α(x ⊗ y) = (αx) ⊗ y = x ⊗ (αy), (x1 ⊗ y1 ) + (x2 ⊗ y2 ) = (x1 + x2 ) ⊗ (y1 + y2 ) − (x1 ⊗ y2 ) − (x2 ⊗ y1 ), as in Section 3.1, for every x, x1 , x2 ∈ X , every y, y1 , y2 ∈ Y, and every α ∈ F (so addition of single tensors is not necessarily a single tensor as bilinear maps are not linear), and yet the collection of all linear combinations of single tensors coincides with the collection of all (finite) sums of single tensors (cf. Theorem 3.3), leading to the notion of tensor product linear space of Section 3.1,

4.4 Additional Propositions

73

X ⊗ Y = T = span R(θ) = span θ(X ×Y) N  = i=1 αi (xi ⊗ yi ) : αi ∈ F , (xi , yi ) ∈ X ×Y, N ∈ N  N = i=1 xi ⊗ yi : (xi , yi ) ∈ X ×Y, N ∈ N .

Now particularize to finite-dimensional spaces. Set X = F m and Y = F n. m Consider the direct sum j=1 F n of m copies of F n. Equivalently, the collection F m (F n ), of all m-tuples of n-vectors, {(y1 , · · · , ym ) : yj ∈ F n , j = 1, . . . , m} =  m n m equipped with the usual direct sum algebra. Thus regard (F ) = j=1 F n  n and F m (F n ) = j=1 F m as different notations for the same (i.e., isomorphic) linear space, which in turn are isomorphic to F mn = F nm. That is, m n F m (F n ) = Fn ∼ F m = F n (F m ). = F mn = F nm ∼ = j=1

j=1

The Kronecker product x  y of two vectors x = (ξ1 , . . . , ξm ) in F m and y = (υ1 , . . . , υn ) in F n is the vector x  y in F m (F n ) given by x  y = (ξ1 y, . . . , ξm y) = (ξ1 υ1 , . . . , ξ1 υn , · · · , ξm υ1 , . . . , ξm υn ). There are vectors in F m (F n ) that are not Kronecker products (e.g., (0, 1, 1, 0) in R 2 (R 2 ) ∼ = R 4 is not a Kronecker product for any pair of vectors (x, y) in 2 2 R × R ), and the sum of Kronecker products is not necessarily a Kronecker product. The next propositions show how the notion of Kronecker product for a pair (x, y) in F m ×F n fits into the abstract notion of tensor products. Proposition 4.A. The operation  is distributive with respect to addition in

F mn and bilaterally homogeneous with respect to scalar multiplication:

α(x  y) = (αx)  y = x  (αy), (x1  y1 ) + (x2  y2 ) = (x1 + x2 )  (y1 + y2 ) − (x1  y2 ) − (x2  y1 ), for every x, x1 , x2 ∈ X , every y, y1 , y2 ∈ Y, and every α ∈ F . Proposition 4.B. The collection F m  F n of all linear combinations of Kronecker products is such that Fm  Fn = =

N



yi ) : αi ∈ F , (xi , yi ) i=1 αi (xi N m n yi : (xi , yi ) ∈ F ×F , i=1 xi



m

n

∈ F ×F , N ∈ N  N ∈N ,



and F m  F n is a linear space over F . The linear space F m  F n is referred to as the Kronecker product space. m Let {ej }m (made up of unit vectors). For j=1 be the canonical basis for F each j = 1, . . . , m take yj = (υ j,1 , . . . , υ j,n ) in F n and set y = (y1 , . . . , ym ) = m (υ1,1 , . . . , υ (F n ) ∼ = F mn, which is naturally identi1,n , · · · , υm,1 , . . . , υm,n ) in F m m n m fied with j=1 ej  yj . Thus F (F ) ⊆ F  F n. The converse identification

74

4. Interpretations

is clear by the definition of Kronecker product: F m  F n ⊆ F mn . These inclusions stand for embeddings so that Fm  Fn ∼ = F m (F n ) ∼ = F mn .

Proposition 4.C. Let · ; ·F mn , · ; ·F m , and · ; ·F n be the usual inner products equipping the linear spaces F mn, F m and F n, respectively. Then x  y ; u  vF mn = x ; uF m y ; vF n

for every

x, u ∈ F m and y, v ∈ F n .

Recall from Remark 4.6 that single tensors as in Corollary 4.5 are given by (x ⊗ y)(u, v) = x ; uF m y ; vF n

for every (x, y), (u, v) ∈ F m ×F n .

So the proposition below shows that Kronecker products x  y in F m  F n are identified with such single tensors x ⊗ y in the tensor product space F m ⊗ F n . Proposition 4.D. The mapping Θ : F m  F n → F m ⊗ F n defined by N  N N Θ xi  yi = xi ; · F m yi ; · F n = xi ⊗ yi i=1

i=1

i=1

is an isomorphism between the linear spaces F m  F n and F m ⊗ F n . Therefore F m  F n is isomorphic to F m ⊗ F n , Fm  Fn ∼ = Fm ⊗ Fn.

Indeed, the natural bilinear map θ  : F m ×F n → F m ⊗ F n of Corollary 4.5 for the particular case of Remark 4.6 is given by θ (x, y)(u, v) = (x ⊗ y)(u, v) = x ; uF m y ; vF n for every (u, v) ∈ F m ×F n , and so the natural bilinear map θ : F m ×F n → F m  F n is given by θ = Θθ  as in Corollary 3.6. Thus consider the tensor product (F m  F n , θ) of the linear spaces F m and F n , and consequently the Kronecker product space is a tensor product space of F m and F n . Since F m  F n has been identified with F m (F n ) we get by transitivity Fm ⊗ Fn ∼ = F m (F n ).

Summing up:

Fm ⊗ Fn ∼ = F mn . = Fm  Fn ∼ = F m (F n ) ∼

Compare the above relations with Proposition 3.F in Section 3.4. Next take arbitrary linear transformations A in L[F m ] and B in L[F n ], and let them be represented with respect to the canonical basis for F m and F n by the square (m×m and n×n) matrices ⎛ ⎞ ⎞ ⎛ α11 . . . α1m β11 . . . β1n ⎜ .. ⎟ and B = ⎜ .. .. ⎟ . A = ⎝ ... ⎝ . . ⎠ . ⎠ αm1 . . . αmm

βn1 . . . βnn

4.4 Additional Propositions

75

The Kronecker product AB of the matrices A and B in L[F m ] in L[F n ] is ⎛ ⎞ α11 B . . . α1m B ⎜ .. ⎟ in Lm F n ∼ L[F mn ], A  B = ⎝ ... = i=1 . ⎠ αm1 B . . . αmm B a square (mn × mn) matrix. Proposition 4.E. For each A ∈ L[F m ] and B ∈ L[F n ],  m  m (A  B) (x  y) = α1,j ξj By, · · · , αm,j ξj By = Ax  By, j=1

j=1

and so (A  B) for every

N i=1

N

i=1

xi  yi =

N i=1

(A  B) (xi  yi ) =

N i=1

Axi  Byi ,

xi  yi ∈ F m  F n .

Consider the notion of tensor product of linear transformations A ⊗ B as in Definition 3.16 and Theorem 3.17. Let Θ be as in Proposition 4.D. Proposition 4.F. For each A ∈ L[F m ] and B ∈ L[F n ], Θ(A  B)Θ−1 = A ⊗ B, where Θ ∈ L[F m  F n, F m ⊗ F n ] is an isomorphism such that Θ(F m  F n ) = F m ⊗ F n . Thus the Kronecker product of matrices is similar to the tensor product of linear transformations, AB ∼ = A ⊗ B. As before, equality of tensor products is regarded up to isomorphism, so that all these identifications, viz., F m  F n ∼ = x ⊗ y and A  B ∼ = = F m ⊗ F n, x  y ∼ ∼ A ⊗ B are usually expressed by writing = for = . So it is also usual to write Fm  Fn = Fm ⊗ Fn,

x  y = x ⊗ y,

A  B = A ⊗ B.

The linear space L[F m, F n ] of all linear transformations of F m into F m is naturally identified with the linear space F m,n of all m×n matrices (representing transformations in L[F m, F ] with respect to the canonical basis for F m and F n), L[F m, F n ] ∼ = F m,n . Proposition 4.G.

Fm ⊗ Fn ∼ = F m,n .

Proposition 4.H. If Pn [0, 1] is the linear space of all polynomials in the variable t ∈ [0, 1] with coefficients in F of degree not greater than n ∈ N ,

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4. Interpretations

  n Pn [0, 1] = p ∈ F [0,1] : p(t) = i=0 αi ti , t ∈ [0, 1], with each αi in F , then for every m, n ∈ N Pn [0, 1] ⊗ Pm [0, 1] ∼ = P(n+1)(m+1) [0, 1]. Let X and Y be finite-dimensional linear spaces over the same field F with dim X = m and dim Y = n so that X ∼ = F n , and X ⊗ Y ∼ = Fm ⊗ Fm = F m, Y ∼ (cf. Corollary 3.11). Let σ(A) denote the spectrum of A ∈ L[X ] (which in this finite-dimensional case coincides with the set of all eigenvalues of A). The trace and the determinant of A ∈ L[X ] are defined as the sum and the product of its eigenvalues (including multiplicity), denoted by tr(A) and det(A). If Λ and Δ are nonempty sets of scalars (i.e., nonempty subsets of F ), then the set product Λ·Δ is defined as the subset of F consisting of the products of all elements from Λ with all elements from Δ. To avoid empty spectrum set F = C . Proposition 4.I. If X and Y are complex finite-dimensional linear spaces with dim X = m and dim Y = n, and if A ∈ L[X ] and B ∈ L[Y], then (a) σ(A ⊗ B) = σ(A) · σ(B), (b) tr(A ⊗ B) = tr(A) · tr(B), (c) det(A ⊗ B) = det(A)m · det(B)n . All the above results dealt with finite-dimensional spaces where tensor products are identified with Kronecker products. The following two results are not restricted to the finite-dimensional case. Proposition 4.J. If— X and Y are infinite-dimensional, then the embedding in Remark 4.9(f ) is proper : X ⊗ Y ⊂ b[X ×Y, F ]. The next proposition extends the claim in the proof of Corollary 4.10.  Proposition 4.K. Let i xi ⊗ yi be an element in the tensor product space of linear spaces X and Y. The following assertions are pairwise equivalent.  (a) i xi ⊗ yi = 0.  (b) i f (xi )g(yi ) = 0 for every f ∈ X and every g ∈ Y .  (c) i f (xi )yi = 0 for every f ∈ X .  (d) i g(yi )xi = 0 for every g ∈ Y . Notes: Grothendieck’s fundamental work in the 1950’s on the metric theory of tensor products has been unified and updated by Diestel, Fourie and Swart in 2008 [17]. See also Pisier’s 2012 exposition [77]. In Grothendieck’s pioneering work, the notion of tensor product space was essentially given in terms of the dual of the linear space of bilinear forms (or, more generally, the linear space of linear maps of bilinear maps). The same sort of tensor product definition also

4.4 Additional Propositions

77

appears in Halmos’s 1958 book on finite-dimensional vector spaces [30, Section 24]. Another representative work along this line (dual of the linear space of bilinear forms) is Ryan’s 2002 book on tensor products of Banach spaces [83]. From the algebraic point of view, this is the approach discussed in Sections 4.2 and 4.3 (and also in Section 4.4) in this chapter. Tensor products of normed spaces and norms on tensor product spaces will be considered in Chapter 7. On the other hand, the alternative (but still usual) approach for defining tensor products as in Section 4.1 relies on quotient spaces of free linear spaces (which are equivalent to the linear space of formal linear combinations) of Cartesian products of linear spaces. Such a quotient space approach has been sometimes referred to as the algebraic tensor product (although the linearbilinear approach is equally algebraic). See, for instance, [5, pp. 22–25], [92, Section 3.4] and [81, Chapter 14] for a linear space version, and [69, Section IX.8] and [66, Section XVI.1]) for a module version. Chapter 4 has dealt with interpretations of the axiomatic approach of Chapter 3, focusing on two distinct lines of action to exhibit concrete examples of tensor product spaces of a pair of linear spaces. Most propositions here in Section 4.4 aim at examples and properties for the finite-dimensional case, which boils down to the notion of Kronecker products. Propositions 4.A to 4.F are standard basic properties of Kronecker products that can be found in many texts dealing with this topic. For instance, see the unified introductory exposition in [48, Section 3]. For Proposition 4.G see, e.g., [5, Problem 2.T]. Proposition 4.H is again an application of Corollary 3.10 and Proposition 1.C (as dim Pn [0, 1] = n + 1; see, e.g., [52, Example 2.M]). The finite-dimensional spectral properties in Proposition 4.I are standard; see, e.g., [94, Proposition 1.10 and its Corollary]. For Proposition 4.J see, e.g., [83, Exercise 1.10 and Proposition 1.2]. Proposition 4.K is an extended version of the claim in the proof of Corollary 4.10 (see, e.g., [83, Proposition 1.2]).

Suggested Readings Brown and Pearcy [5] Defant and Floret [15] Greub [27] Halmos [30] Jarchow [42]

Lang [66] MacLane and Birkhoff [69] Roman [81] Ryan [83] Yokonuma [94]

5 Normed-Space Results

This chapter outlines the essential normed-space notions required in the book. It focuses on bounded (i.e., continuous) linear transformations and normed quotient spaces. As in Chapter 1, the purpose here is to put together only those results necessary in the forthcoming chapters. Normed-space aspects of bilinear maps will be discussed in Chapter 6. Chapters 5 and 6 enable us to advance an axiomatic theory of tensor products of Banach spaces.

5.1 Notation, Terminology and Definitions Again, all linear spaces are over the same field F . A normed space is a linear space X equipped with a norm  ·  : X → R . A norm generates a metric, and so normed spaces are metric spaces. A normed space X may also be denoted by (X ,  · ), and a subscript may be inserted if necessary, for instance  · X . Linear manifold. By a linear manifold M of a normed space X we mean a linear manifold of the linear space X which inherits its norm from the normed space X ; the norm on the normed space (M,  · M ) is the restriction of the norm on (X ,  · X ), that is,  · M =  · X |M : M → R , which is indeed a norm on M. A subspace of a normed space is a closed linear manifold (closed with respect to the metric generated by the norm). Every finite-dimensional linear manifold of any normed space is a subspace. The closure M− of a linear manifold M of a normed space is a subspace. A Banach space is a complete normed space (complete with respect to the metric generated by the norm). Subspaces (closed linear manifolds) of Banach spaces are Banach spaces. A Hilbert space is a Banach space whose norm is induced by an inner product. Topological isomorphism. In this context, a homeomorphism is an invertible map between metric spaces which is continuous and has a continuous inverse. A topological isomorphism is an isomorphism between normed spaces which is also a homeomorphism, that is, an invertible continuous linear transformation between normed spaces whose inverse is continuous. If there exists a topological isomorphism of a normed space X onto a normed space Y, then we say X and Y are topologically isomorphic — same notation: X ∼ = Y. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Kubrusly, Bilinear Maps and Tensor Products in Operator Theory, Universitext, https://doi.org/10.1007/978-3-031-34093-2 5

79

80

5. Normed-Space Results

Isometric isomorphism. This is a particular case of a topological isomorphism that preserves distance. An isometry between metric spaces is a map that preserves distance. An isometric isomorphism is a surjective linear isometry or, equivalently, an invertible linear isometry. (Recall: (i) the inverse of an invertible linear transformation is linear, (ii) the inverse of an invertible isometry is an isometry, (iii) an isometry is always injective, (iv) the term isometry in this context will from now on mean linear isometry: a linear transformation between normed spaces that preserves distance.) If there exists an isometric isomorphism of a normed space X onto a normed space Y then we say that X and Y are isometrically isomorphic — still the same notation X ∼ = Y. Topology on Cartesian products and direct sums. Take a pair, say (X ,  · X ) and (Y,  · Y ), of normed spaces and let X ×Y be the Cartesian product of the linear spaces X and Y. Consider the following equivalent alternatives (which naturally extend to any countable family of normed spaces). (i) Either equip the Cartesian product X ×Y with any product metric d generated by those norms, say, dp for p ≥ 1 or d∞ , from (X ×Y)×(X ×Y) to R , given by dp ((x, y), (x , y  )) = (x − x pX +y − y  pY )1/p and d∞ ((x, y), (x , y  )) = max{x − x X , y − y  Y } for (x, y), (x , y  ) ∈ X ×Y, and this leads to a metric space (X ×Y, d) where the set X ×Y is equipped with a product metric d; (ii) or equip the direct sum X ⊕ Y (a linear space whose underlying set is X ×Y) with any of its standard norms  · , say, (x, y)p = (xpX + ypY )1/p for p ≥ 1 or (x, y)∞ = max{xX , yY } for (x, y) ∈ X ⊕ Y, and this leads to a normed space (X ⊕ Y,  · ) = (X ×Y, ⊕,  , F ,  · ), where the norm  ·  on X ⊕ Y generates a product metric d on the Cartesian product X ×Y. Bounded linear transformation. If (X ,  · X ) and (Y,  · Y ) are normed spaces, then let B[X , Y] denote the linear space of all bounded linear transformations of X into Y, equivalently, of all continuous linear transformations of X into Y, with respect to the norms  · X and  · Y . The kernel N (T ) of a bounded linear transformation T ∈ B[X , Y] is a subspace of X and the range R(T ) is a linear manifold of Y. The induced uniform norm  · B[X ,Y] of xY T ∈ B[X , Y] is a norm on B[X , Y] defined by T B[X ,Y] = supx=0 T xX , and so T xY ≤ T B[X ,Y] xX for every x ∈ X (i.e., T is a bounded transformation). When equipped with this norm, the normed space (B[X , Y],  · B[X ,Y] ) becomes a Banach space if (Y,  · Y ) is a Banach space (the converse holds if X = {0}). We will drop subscripts from norms whenever this is clear in the context and will write x, T x, T  all with the same norm notation. An isometry is a transformation T ∈ B[X , Y] such that T x = x for every x ∈ X , which implies T  = 1. Thus T is an isometric isomorphism if and only if it is surjective and T x = x for every x ∈ X . Moreover, T ∈ B[X , Y] is a contraction if T x ≤ x for every x ∈ X , which means T  ≤ 1. By an operator we mean a bounded linear transformation of a normed space into itself: T is an operator if T ∈ B[X , X ]. We use the notation B[X ] as a short for the normed algebra B[X , X ] of all operators on X (so B[X ] is a Banach algebra if and only

5.1 Notation, Terminology and Definitions

81

if X is a Banach space). If S and T lie in the normed algebra B[X ], then their product S T (i.e., composition S ◦ T ) also lies in B[X ] and ST  ≤ ST . Bounded linear functionals, dual and normed-space adjoint. It is clear that B[X , Y] ⊆ L[X , Y]. Let X ∗ = B[X , F ] ⊆ L[X , F ] = X  be the dual of the normed space X , consisting of all linear functionals f : X → F from X  which are continuous (i.e., bounded) with respect to the norm  · X on X and the usual norm | · | on F . Since (F , | · |) is a Banach space, it follows that the dual X ∗ = (X ,  · X )∗ = (X ∗,  · X ∗ ) = (B[X , F ],  · B[X ,F ] ) of any normed space X is a Banach space, where the norm on X ∗ is the induced uniform norm on (x)| ∗ B[X , F ] which is given by f X ∗ = f B[X ,F ] = supx=0 |f xX for every f ∈ X . ∗ ∗ ∗ So |f (x)| ≤ f X ∗ xX for every x ∈ X . Let T ∈ B[Y , X ] be the normedspace adjoint of T ∈ B[X , Y], which is the unique transformation in B[Y ∗, X ∗ ] for which T ∗g = g T ∈ X ∗ for every g ∈ Y ∗ (i.e., (T ∗g)(x) = g(T x) ∈ F for every g ∈ Y ∗ and every x ∈ X — this is also written as x ; T ∗ g = T x ; g ). The usual basic properties of Hilbert-space adjoints are transferred to normedspace adjoints. For instance, T ∗  = T , (T + S)∗ = T ∗ + S ∗ for S ∈ B[X , Y] and, slightly different, (αT )∗ = αT ∗ for α ∈ F . If S, T ∈ B[X ], then (T S)∗ = S ∗ T ∗, and I ∗ is the identity in B[X ∗ ] where I is the identity in B[X ]. (See, e.g., [70, Section 3.1] or [87, Section 3.2].) When a norm either is clear or is immaterial, we refer to the normed spaces X , B[X , Y], B[X ] or X ∗ without specifying which norm is being considered, and write norms without subscripts, such as x T  = sup T x = sup T x = sup T x = sup T x, x=0

f  = sup x=0

|f (x)| x

x=1

x≤1

x 1. For |β| = 1 take a2 + b2 = 1 with ab β > 0 so that (a + βb)2 = a2 + b2 + 2abβ = 1 + 2abβ > 1. γ2 2 For 0 < |β| < 1 take a2 + b2 = 1 with b > 0, a β > 0 and 1+γ 2 < a < 1 where 2

2 γ = 1−β 2|β| > 1. This implies (a + βb) > 1. Indeed,

5.2 Extension of Bounded Linear Transformations

a2 >

γ2 1+γ 2

89

⇐⇒ a2 (1 + γ 2 ) > γ 2 =⇒ a2 > γ 2 − a2 γ 2 = γ 2 (1 − a2 ) = γ 2 b2

=⇒ |a| > γb =

1−β 2 2|β| b =⇒ 2 2 2

2|aβ| > b(1 − β 2 ) ⇐⇒ 2aβ > b(1 − β 2 ) =⇒

(a + βb)2 = a + β b + 2abβ = a2 + b2 − b2 + β 2 b2 + 2abβ

= 1 − b2 (1 − β 2 ) + 2abβ = 1 + b 2aβ − b(1 − β 2 ) > 1. (g) Generalizations of the Hahn–Banach Theorem. If we could generalize the Hahn–Banach Theorem by replacing the one-dimensional Banach space F (the codomain of f ) by an arbitrary Banach space Y of arbitrary dimension, then we would get a full Banach-space version of the purely algebraic result in item (a), now dealing with a bounded linear extension T : X → Y of a bounded linear transformation T : M → Y. But this fails in general. Such an extension T : X → Y may not exist without extra assumptions (see, e.g., [11]). Extra assumptions may be placed on each of the four players involved, viz., the normed space X , the linear manifold M, the bounded linear transformation T , and the Banach space Y — for instance, it holds in (d) for dim Y = 1. (Note: there is no Hahn–Banach Theorem for bilinear forms, as will be seen in Chapter 6.) (h) A classical approach. According to Nachbin [71], a Banach space Y has the extension property if for every Banach space X and every subspace M of X, every bounded linear transformation T : M →Y has a bounded linear extension T : X →Y over X with the same norm. By the Hahn–Banach Theorem, a onedimensional Banach space has the extension property, but this is not a property of every Banach space (see also [11]): there is no Hahn–Banach Theorem for an arbitrary bounded linear transformation into an arbitrary Banach space. (i) The complementation approach. If we look at the normed space X instead, or at the lattice of all its linear manifolds M, then a Banach-space version of the algebraic result in item (a) for bounded linear transformations can be obtained by restricting M to be complemented in X . This will be considered in Theorem 5.4 below, and it shows again that there is no Hahn–Banach Theorem for arbitrary bounded linear transformations between arbitrary Banach spaces. Complementation will play an important role from now on. Remark 5.3. Complementary Subspaces and Continuous Projection. (a) Recall: (1) every linear manifold of any linear space has an algebraic complement; (2) a subspace of a normed space is complemented if it has a subspace as an algebraic complement — in this case it is called a complemented subspace; (3) if two subspaces are complements of each other, then they are called complementary subspaces; (4) not all subspaces of an arbitrary Banach space are complemented; (5) a normed space is complemented if every subspace is complemented; and (6) if a Banach space is complemented, then it is isomorphic (topologically isomorphic, by the Inverse Mapping Theorem) to a Hilbert space — thus complemented Banach spaces are identified with Hilbert spaces: only Hilbert spaces (up to isomorphism) are complemented [67] (see also [45]).

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5. Normed-Space Results

(b) If X is a normed space and E : X → X is a continuous projection, then R(E) and N (E) are complementary subspaces of X . This is immediate since R(E) and N (E) are complementary linear manifolds and R(E) = N (I − E). Conversely, if M and N are complementary subspaces of a Banach space X , then the (unique) projection E : X → X with R(E) = M and N (E) = N is continuous (since on a Banach space a projection with closed range and closed kernel is continuous). (See, e.g., [12, Section III.13], [70, Section 3.2], [52, Problem 435], [56, Section A.2].) Thus if X is a Banach space, then the assertions below are equivalent. (i) A subspace M of X is complemented . (ii) There exists a continuous projection E : X → X with R(E) = M. Theorem 5.4. Let M be a linear manifold of a Banach space X . (a) Every bounded linear transformation T : M → Y into an arbitrary Banach space Y has a bounded linear extension T : X → Y over X if and only if the closure M− of M is complemented in X . (b) If M− = R(E) for a projection E ∈ B[X ] with E = 1, then T = T . Proof. (a) Let M be a linear manifold of a Banach space X so that the (closed) subspace M− of X is itself a Banach space. Take an arbitrary Banach space Y and an arbitrary T ∈ B[M, Y]. The following assertions are equivalent. (a1 ) For every Y and every T ∈ B[M,Y] there is T ∈ B[X ,Y] such that T|M = T . (a2 ) M− is complemented in X . (a1 ) ⇒(a2 ). Consider an arbitrary linear manifold M of the Banach space X and take the Banach space M−. If (a1 ) holds, then it holds in particular for T : M → M ⊆ M− defined by T u = u for every u ∈ M. Since T acts as the identity on M, then it is trivially linear and continuous (i.e., T ∈ B[M, M− ]). If (a1 ) holds, then there exists a linear and continuous extension T : X → M− of T over X (i.e., there exists T ∈ B[X , M− ] such that T|M = T ). Since T|M acts as the identity on M and T|M is linear and continuous, then T|M extends by continuity (cf. Remark 5.2(b)) over M− so that T|M− ∈ B[M−, M− ] is the identity operator on M− and R(T) = M− (as M− = R(T|M−) ⊆ R(T) ⊆ M−). Then the bounded linear T : X → M− ⊆ X acts as the identity on its range, and so it is idempotent. Thus T ∈ B[X , X ] is a continuous projection with range M−. But the range of a continuous projection is a complemented subspace (cf. Remark 5.3). So M− is complemented. (a2 ) ⇒(a1 ). Conversely, suppose M− is complemented. Thus M− is the range of a continuous projection (cf. Remark 5.3). Then let E ∈ B[X ] be a projection such that R(E) = M−. Let Y be an arbitrary Banach space and let T be an arbitrary transformation in B[M, Y]. Extend T by continuity (as in Remark 5.2(b)) over M− to get T ∈ B[M−, Y] for which T|M = T . Since E acts as the

5.2 Extension of Bounded Linear Transformations

91

identity on R(E) = M−, define T = TE ∈ B[X , Y] so that T|M = TE|M = T|M = T . Thus T is a bounded linear extension of T over X . (b) Moreover, under the assumption in (a2 ), T = T|M and T = TE. Hence T  = T|M  ≤ T = TE ≤ TE. Also T = T  since T extends T by continuity. So T  ≤ T ≤ T E. Thus E = 1 implies T = T .   Remark 5.5. Examples Related to Theorem 5.4. (a) Extensions of the identity. Set Y = X . Take T ∈ B[M, Y] such that T acts as the identity on M. That is, T naturally embeds M ⊆ X into X = Y, so that T u = u for every u ∈ M and hence T  = 1. If M is complemented in X , then every continuous projection E in B[X ] with R(E) = M is a continuous extension of T over Y = X (cf. Remark 5.3). The norms, however, are not preserved for projections with norms different from 1. (b) Norms of continuous projections. If E is a nonzero projection in B[X ] (i.e., a continuous projection on a normed space X ), then E = E 2 implies E ≥ 1. Every continuous projection on X has norm greater than or equal to 1 (and every orthogonal projection on an inner product space has norm 1). For instance, E(x1 , x2 ) = (0, x1 + x2 ) and P (x1 , x2 ) = (0, x2 ) for (x1 , x2 ) ∈ R 2   define continuous projections E = 01 01 and P = 00 01 on the Euclidean space R 2 both with R(P ) = {0} ⊕ R ⊆ R ⊕ R = R 2 , such √ the same range, R(E) = 2 that E = 2 and P  = 1, when R is equipped with the Euclidean norm. (Incidentally, for bounded linear transformations A and B, if 0 = A = AB, then B ≥ 1 but not necessarily B = 1 — as there exists E = E 2  > 1.) (c) The converse of Theorem 5.4(b) may fail. Set Y = X (same underlying set, same norm). Take a projection E ∈ B[X ] so that E = E 2 and hence E ≥ 1. Suppose E > 1. Set M = R(E) and T = E|R(E) : R(E) → R(E) ⊆ X = Y, acting as the identity IM on M = R(E). Set T = IX : X → X = Y, the identity on X , and so T|M = IX |R(E) = E|R(E) = T with T  = E|R(E)  = IM  = IX  = T. Thus T = T  and M− = R(E) for a projection E ∈ B[X ] with E = 1. Does there exist another projection P ∈ B[X ] with P  = 1 for which the same M is such that M− = R(P )? (See Corollary 5.6 below.) (d) Nonuniqueness of an extension. Set Y = X = R 2 (the Euclidean space), √  consider the projection E = 01 01 : R 2 → R 2 with E = 2 and take the subspace M = R(E) = {(x1 , x2 ) : x1 = 0} = {0} ⊕ R of R ⊕ R = R 2 = X . Set T = E|R(E) : R(E) → R(E) ⊆ X = Y so that T = E|M ∈ B[M, Y] acts as the identity on R(E). Hence T  = E|R(E)  = 1. Then T1 = E in B[X , Y] is the obvi ous extension of T ∈ B[M, Y] but T1  = T . Now take I = 10 01 : R 2 → R 2, the identity on X . Thus T2 = I in B[X , Y] is another trivial extension of  T ∈ B[M, Y], this time with T2  = T . Next take P = 0 0 : R 2 → R 2, 0 1

92

5. Normed-Space Results

a nontrivial orthogonal projection in B[R 2 ]. Then P  = 1 with R(P ) = R(E) = M so that P |M = E|M ∈ B[M, X ], and T3 = P in B[X , Y] is still another trivial extension of T ∈ B[M, Y], also with T3  = T . Even with norm preservation a continuous extension is not unique. The following consequence of Theorem 5.4 is crucial. Corollary 5.6. Every bounded linear transformation T : M → Y of an arbitrary linear manifold M of any Hilbert space X into an arbitrary Banach space Y has a bounded linear extension T : X → Y over X such that T = T . Proof. (a) Let X be a Hilbert space. By the Projection Theorem, M + M⊥ = X for every subspace M of X , where M⊥ is the orthogonal complement of M, which is a subspace of X such that M ∩ M⊥ = {0} (see Proposition 5.Q in Section 5.4). So every subspace of a Hilbert space is complemented , and Theorem 5.4 applies to every linear manifold M of every Hilbert space X . (b) If X is Hilbert, then we may take the orthogonal projection E onto M− so that E = 1. Thus T = T  by Theorem 5.4(b) (cf. Proposition 5.R).   The examples in Remark 5.5(d) show that even on a Hilbert space and under the norm identity T = T  the extension T of T is not unique. Now let X be a normed space and define the following classes of operators.

 ΓR [X ] = S ∈ B[X ] : R(S)− is a complemented subspace of X ,

 ΓN [X ] = S ∈ B[X ] : N (S) is a complemented subspace of X , Γ [X ] = ΓR [X ] ∩ ΓN [X ],

SΓ [X ] = ΓR [X ] ∪ ΓN [X ].

If X is a Banach space, then the set Γ [X ] = ΓR [X ] ∩ ΓN [X ] is algebraically and topologically large in the sense that it includes nonempty open groups. For instance, the group of all invertible operators from B[X ] (i.e., of all operators from B[X ] with a bounded inverse) is open in B[X ] and trivially included in Γ [X ]. Actually, the class of all compact perturbations of invertible (more generally, of semi-Fredholm) operators lies in Γ [X ] (see, e.g., [55, Remark 5.1]). Corollary 5.7. Let X be a Banach space. Let M be a linear manifold of X for which there is an operator S ∈ SΓ [X ] such that M− = R(S)− or M− = N (S), according to whether S lies in ΓR [X ] or in ΓN [X ], respectively. If T : M → Y is any bounded linear transformation into an arbitrary Banach space Y, then there exists a bounded linear extension T : X → Y of T over X . Proof. Straightforward from Theorem 5.4. If S ∈ SΓ [X ], then either R(S)− or N (S) is complemented. If M− = R(S)− or M− = N (S), then T ∈ B[M, Y] for any Banach space Y has an extension T ∈ B[X , Y] by Theorem 5.4.  

5.3 Normed Quotient Space

93

Take E ∈ B[X ] and T ∈ B[M, Y], where X and Y are Banach spaces and M is a linear manifold of X . Suppose M = R(E). Recall from Remark 5.3 that continuous projections lie in Γ [X ]. Thus if E is a projection, then Corollary 5.7 says that there exists T ∈ B[X , Y] such that T|M = T . However (cf. proof of Theorem 5.4), if L ∈ B[X , Y] is defined by L = T E, then L is an extension of T . In fact, since L|R(E) = T (E|R(E) ) and E is idempotent (so that E|R(E) = IR(E) , identity on R(E)), we get L|R(E) = T . Conclusion: if T ∈ B[X , Y] is an arbitrary extension of T ∈ B[M, Y], where M = R(E) for some projection E ∈ B[X ], then T|R(E) = T and therefore TE = T E = L.

5.3 Normed Quotient Space We have agreed that by a subspace we mean a closed linear manifold of a normed  space. Let M be a linear manifold of a normed space X . Consider the map  ·  : X /M → R defined for each coset [x] in the quotient space X /M by     [x]  = x + M = inf x + u = d(x, M), u∈M

the distance of x to M, which is invariant for all representatives in [x].   Theorem 5.8. If M is closed, then  ·  : X /M → R is a norm on X/M.     Proof. Let M be a linear manifold of a normed   space (X ,  · ). The map ·   defines a function from X /M to R since [x] depends only on the coset [x] and not on a particular representative x in [x]. As for nonnegativeness,   [x]  ≥ 0 for every [x] ∈ X /M trivially. Absolute homogeneity and subadditivity are easily verified. In fact, α[x] = [αx]

and

[x] + [y] = [x + y]

for every [x], [y] in X /M and every scalar α. Since M is a linear manifold,     α[x]  = [αx]  = inf αx + u = inf αx + αu u∈M u∈M   = |α| inf x + u = |α| [x] , u∈M

    [x] + [y]  = [x + y]  = inf x + y + u = inf x + u + y + u u∈M u∈M      ≤ inf x + u + inf y + u = [x]  + [y] , u∈M

u∈M

  for every [x], [y] in X /M and every scalar α. Thus the map  ·  : X /M → R is a semi-norm for every linear manifoldM of X . Now suppose M is closed in X , and take an arbitrary x ∈ [x]. If [x]  = 0, then d(x, M) = 0 and so

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5. Normed-Space Results

x ∈ M− = M. This means x ∈ [0] (since M is the origin [0] of X /M — see Section 1.3), and hence [x] = [0]. Equivalently,   [x]  > 0 whenever [x] = 0.   If M is a subspace of a normed space X , then equip the quotient space X /M with this norm, which is referred to as the quotient norm. When we refer to the normed space X /M, it is understood that M is a subspace and the quotient space X /M is equipped with the quotient norm. If M is a subspace so that X /M is a normed space, then consider the natural quotient map π : X → X /M of Section 1.3, which is given by π(x) = [x] ∈ X /M for every x ∈ X (and is linear and surjective   as we saw in Section 1.3). This π is a contraction. Actually, π(x) = [x]  = d(x, M) = inf u∈M x + u and hence   π(x) ≤ x for every x ∈ X so that π ∈ B[X , X /M]. More is true. First note the trivial cases.   M = {0} ⇐⇒ π(x) = inf x + u = x ∀ x ∈ X . u∈{0}

In fact, according to Section 1.3, M = {0} ⇐⇒ X /M = X /{0} ∼ = X ⇐⇒ π ∼ = I. Also π is an isometry ⇐⇒ π is injective (since π is injective ⇐⇒ M = {0}, as we have also seen in Section 1.3). Moreover (compare with Section 1.3 again),   M = X ⇐⇒ π(x) = inf x + u = 0 ∀ x ∈ X ⇐⇒ π = 0. u∈X

If M is a proper subspace of X , then the contraction π has norm one: M = X ⇐⇒ π = 1.

  Indeed, for M = {0} (otherwise π is an isometry), π = supx=1 π(x) = supx=1 d(x, M) = 1 by the Riesz Lemma (see, e.g., [52, Lemma 4.33]). Remark 5.9. The Three-Space Property. Recall that a linear manifold of a Banach space is itself a Banach space if and only if it is closed. If M is a (closed) subspace of a Banach space X , then the quotient space X /M is a Banach space (this an important and often used result — see, e.g., [52, Proposition 4.10], [70, Theorem 1.7.7], [12, Theorem III.4.2(b)], [87, Theorem 3.13]). Conversely, if M is a Banach space (i.e., M is complete in a normed space X so that M is closed) and the quotient space X /M is a Banach space, then X is a Banach space (see, e.g., [52, Problem 4.13], [12, Exercise III.4.5]). If X /M is a normed space equipped with the quotient norm, then M must be a subspace of X ; if X is Banach, then so is M. Thus if any two of M, X and X /M are Banach spaces, then so is the other (see also [70, Theorem 1.7.9]). Theorem 1.4 on linear transformations can be extended to bounded linear transformations when quotient spaces are defined for normed spaces, now with ∼ = standing for topologically isomorphic rather than (algebraically) isomorphic.

5.3 Normed Quotient Space

95

Theorem 5.10. Let X and Z be normed spaces over the same field, let M be a subspace of X , consider the normed quotient space X /M, and take the natural quotient map π : X → X /M. (a) (Universal Property): If T ∈ B[X , Z] and M ⊆ N (T ), then there exists a unique T ∈ B[X /M, Z] such that T = T ◦ π. In other words, the diagram T

X −−−→ 

π



Z  ⏐ ⏐T X /M

commutes. Moreover, N (T) = N (T )/M, R(T) = R(T ), and T = T . (b) (The First Isomorphism Theorem in Banach Space): If X and Z are Banach spaces, T ∈ B[X , Z], and R(T ) is closed, then R(T ) ∼ = X /N (T ). Proof. (a) Let X , Z be normed spaces, M a subspace of X , and π : X → X /M the quotient map. Suppose the linear transformation in Theorem 1.4 is continuous. In other words, replace L ∈ L[X , Z] with T ∈ B[X , Z] in Theorem 1.4, and recall that now π is a contraction, thus continuous. We know from Theorem 1.4 that, if M ⊆ N (T ), then there is a unique linear transformation T : X /M → Z such that T = T ◦ π, that is, T[x] = T x for every x ∈ X . Claim. T ∈ B[X /M, Z]. Proof of Claim. If the linear transformation T : X /M → Z is not continuous, which means it is not continuous at the origin [0], then there exists an X /Mvalued sequence {[xn ]} such that [xn ] → [0] in X /M

while

T xn = T[xn ] → T[0] = T 0 = 0 in Z.

Take an arbitrary ε > 0. Since  [xn ] → [0] = M ⊆  N (T ), then there exists a positive integer nε such that [xn ] − [0]  = [xn ]  = inf u∈M xn − u < ε for every n ≥ nε . Therefore there exists an M-valued sequence {u(ε)n } for which xn − u(ε)n  < ε whenever n ≥ nε . Then, since M ⊆ N (T ), for every n ≥ nε T xn  = T xn − T u(ε)n  = T (xn − u(ε)n ) ≤ T  xn − u(ε)n  < T  ε. Hence T xn → 0, which is a contradiction. Thus T is continuous.

 

Also by Theorem 1.4 we get N (T) = N (T )/M and R(T) = R(T ). Moreover, since {0} ⊆ [0] = M so that X \M ⊆ X \{0}, and since T = T ◦ π, T = sup

[x]=[0]

T[x]  [x] 

= sup

x∈M

T x  [x] 

≤ sup

x∈{0}

T x  [x] 

= T  = T ◦ π ≤ T π ≤ T

because π ≤ 1 (actually, π = 1). Therefore T = T .

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5. Normed-Space Results

(b) Since T ∈ L[X , Z] between linear spaces, then Theorem 1.4(b) ensures that X /N (T ) is algebraically isomorphic to R(T ), and (cf. proof of Theorem 1.4(b)) T : X /N (T ) → R(T ) is an isomorphism between them. However, since X and Z are normed spaces and T ∈ B[X , Z], the linear manifold N (T ) is closed in X (i.e., it is a subspace of X ). Thus if X is a Banach space, then so is X /N (T ) (cf. Remark 5.9). Moreover, if Z is a Banach space as well and R(T ) is closed in Z, then R(T ) is itself a Banach space. So T ∈ B[X /N (T ), R(T )] is injective and surjective between Banach spaces and therefore, by the Inverse Mapping Theorem, its inverse is continuous (cf. Proposition 5.B in Section   5.4), so that T is a topological isomorphism of X /N (T ) onto R(T ). Remark 5.11. The Dual of the First Isomorphism Theorem. If X and Z are Banach spaces, T ∈ B[X , Z], and R(T ) is closed, then R(T ) ∼ = X /N (T ), where ∼ = means topologically isomorphic. This is the First Isomorphism Theorem in Banach Space (cf. Theorem 5.10(b)). Its dual reads as follows: if X and Z are normed spaces, T ∗ ∈ B[Z ∗, X ∗ ] is the normed-space adjoint of T ∈ B[X , Z], R(T ) is closed, and (Z/R(T ))∗ is the dual of Z/R(T ), then (see, e.g., [70, Theorems 1.10.17, Lemma 3.1.16])  ∗ N (T ∗ ) ∼ = Z/R(T ) , where in this case ∼ = means isometrically isomorphic. Definition 5.12. Let X and Y be nonzero normed spaces. A transformation Q ∈ B[X , Y] is called a quotient transformation if it is surjective and y =

inf

x∈Q−1 ({y})

x

for every

y ∈ Y.

Equivalently, Q is a quotient transformation if the following condition holds. Take an arbitrary y ∈ Y. Then y ≤ x for all x ∈ Q−1 ({y}), and for every ε > 0 there exists x ∈ Q−1 ({y}) such that x ≤ y + ε. In particular, for y = 0 and for every ε > 0, set ε = ε y so that x ≤ (1 + ε )y. If there is a quotient transformation of X onto Y, then Y is a quotient of X. Theorem 5.13. Let X and Y be normed spaces. (a) If Q : X → Y is a linear surjective contraction and for each y ∈ Y there is an x ∈ Q−1({y}) such that y = x, then Q is a quotient transformation. From now on suppose Q ∈ B[X , Y]. (b) Q is an injective quotient transformation if and only if Q is an isometric isomorphism. (So (1) if Q is a quotient transformation, then it is injective if and only if it is an isometric isomorphism, (2) every isometric isomorphism is an injective quotient transformation, and (3) if two normed spaces are isometrically isomorphic, then one is a quotient of the other.)

5.3 Normed Quotient Space

97

(c) If Q is a quotient transformation, then (c1 ) Qx ≤ x every x ∈ X

(i.e., Q is a contraction),

(c2 ) for every x ∈ X and every ε > 0 there exists an xε ∈ Q−1 ({Qx}) ⊆ X for which Qx = Qxε and xε  < Qx + ε. (d) Q is a quotient transformation if and only if Q maps the open unit ball in X onto the open unit ball in Y. (Thus Q = O.) (e) If Q is a quotient transformation, then (e1 ) Qis an open map (i.e., it takes open sets into open sets), (e2 ) Q = 1. (f) If M is a proper subspace of X (i.e., M = X ), then the quotient map π : X → X/M is a quotient transformation. (So X/M is a quotient of X .) Let ∼ = stand for isometrically isomorphic.  ∈ B[X, Y]  (g) Let X , X, Y, Y be normed spaces. Suppose Q ∈ B[X , Y] and Q ∼  are isometrically isomorphic (i.e., Q = Q). Then Q is a quotient transfor is a quotient transformation. mation if and only if Q (h) If Q = O and Y ∼ = X /N (Q), then Q is a quotient transformation. (i) If X and Y are Banach spaces, then the converse holds: if Q is a quotient transformation then Y ∼ = X /N (Q). (j) If Q ∈ B[X , Y] is a quotient transformation, then so is its extension over  ∈ B[X, Y].  completion Q Proof. Let Q : X → Y be linear and surjective between normed spaces. (a) Suppose Q is a contraction (i.e., Qx ≤ x for every x ∈ X so that Q ∈ B[X , Y]). Since Q is surjective, being a contraction means y ≤ x for every y ∈ Y and every x ∈ Q−1 ({y}) ⊆ X . If, in addition, for every y ∈ Y = R(Q) there exists an xy ∈ Q−1 ({y}) ⊆ X such that y = xy , then Q is a quotient transformation by Definition 5.12, which proves (a). Indeed, xy  = y ≤

inf

x∈Q−1 ({y})

x

=⇒

y =

inf

x∈Q−1 ({y})

x.

From now on suppose Q ∈ B[X , Y]. According to Definition 5.12, Q is a quotient transformation if and only if Q is surjective and Qx =

inf

z∈Q−1 ({Qx})

z

for every

x ∈ X.

(b) Suppose Q is a quotient transformation. Then it is surjective. If it is injective, then it is invertible and so Q−1 ({Qx}) = {x}, which implies

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Qx = inf z∈{x} z = x, for every x ∈ X . Thus Q is a surjective isometry, equivalently, an invertible isometry, that is, an isometric isomorphism. Conversely, every invertible isometry satisfies the above displayed identity. (c) Let Q be a quotient transformation. (c1 ) As the inclusion A ⊆ F −1 (F (A)) holds for every function F : X → Y and for every set A ⊆ X , it follows that Qx = inf z∈Q−1 (Q({x})) z ≤ inf z∈{x} z = x for every x ∈ X . (c2 ) Take an arbitrary x ∈ X and an arbitrary ε > 0. Set y = Qx. Since y = inf x∈Q−1 ({y}) x, there exists an xε ∈ Q−1 ({y}) such that xε  < y + ε = Qx + ε. But Q(Q−1 ({y})) = {y} because Q is surjective. So Q(xε ) = y = Qx. (d) Let BX and BY be open unit balls in X and Y (centered at the origin). Part I. Let Q be a quotient transformation. If x ∈ BX , then Qx ≤ x < 1

by (c1 ). Hence Q(BX ) ⊆ BY . Conversely, take any y ∈ BY . So y < 1. As Q is surjective, take any x ∈ X such that Qx = y. Take ε ∈ (0, 1 − y). By (c2 ) there is an xε ∈ X for which Qxε = y = Qx and xε  < y + ε < 1. Thus xε ∈ BX and so y ∈ Q(BX ). Hence BY ⊆ Q(BX ). Then Q(BX ) = BY .

Part II. Conversely, suppose Q ∈ B[X , Y] is such that Q(BX ) = BY . Thus Q

is surjective because BY ⊂ R(Q) ⊆ Y implies R(Q) = Y. Since Q(BX ) ⊆ BY ,

 x ∈ BX =⇒ Qx ≤ 1 =⇒ Q = sup Qx ≤ 1. x∈BX

Since every y ∈ BY = Q(BX ) is such that y = Qx for some x ∈ BX , then for every ε > 0 there exists xε ∈ BX for which Qxε  = 1 − ε. Hence Q = sup Qx = 1. x∈BX

Take an arbitrary x ∈ X . Recall that x ∈ Q−1 ({Qx}) (indeed, as we saw above A ⊆ F −1 (F (A))). Since Q ≤ 1 we get Qx ≤ x. Thus Qx ≤

inf

x∈Q−1 ({Qx})

x

for every

x ∈ X.

On the other hand, take an arbitrary ε > 0. Since Q = 1, there is an xε ∈ BX such that Qxε  + ε > 1 = xε . Again xε ∈ Q−1 (Q({xε })). So for every ε > 0 Qxε  + ε > xε 

for some

xε ∈ Q−1 ({yε }).

Therefore Qx = inf x∈Q−1 ({Qx}) x for every x ∈ X . (e) Let Q be a quotient transformation. (e1 ) Since Q sends the open unit ball in X onto the open unit ball in Y according to (d), then it sends every open ball in X onto an open ball in Y, and therefore it sends every open set in X into an open set in Y, since the open balls are a topological basis for open sets, which precisely means Q is an open map. (e2 ) We have already shown in the proof of (d) that a linear transformation that sends the open unit ball in X onto the open unit ball in Y has norm one.

5.3 Normed Quotient Space

99

(f) Let M be a subspace of the normed space X and consider the natural quotient map π : X → X /M which is given by π(x) = [x] = x + M. Recall from Section 1.3: X /M = {[0]} ⇐⇒ M = X ⇐⇒ π = 0. Observe that π −1 ({π(x)}) = x + M

for every

x ∈ X,

since π −1 (π(S)) = S + M for every subset S of X (cf. Section 1.3). Therefore     inf z = inf z = inf x + u = [x]  = π(x). −1 z∈π

({π(x)})

z∈x+M

u∈M

So the surjective bounded linear nonzero map π is a quotient transformation. (g) Let ∼ = stand for isometrically isomorphic. Take arbitrary transformations  in B[X, Y].  To say that Q ∼ Q in B[X , Y] and Q = Q means (1) X = X(X ) and Y = Y (Y) for some invertible linear isometries (i.e., isometric isomorphisms)  and also (2) Y Q = QX.  X : X → X and Y : Y → Y, Take the open unit balls BX ⊆ X and BY ⊆ Y. Since X and Y are isometric isomorphisms, the open unit balls BX ⊆ X and BY ⊆ Y are of the form BX = XBX and BY = Y BY .  is Therefore, according to (d), Q is a quotient transformation if and only if Q a quotient transformation, because −1    = B . QBX = BY ⇐⇒ Y −1 QXX BX = Y −1 BY ⇐⇒ QB X Y

(h) Take a nonzero Q ∈ B[X , Y]. Since Q is nonzero, linear and continuous, then N (Q) is a proper subspace of X (i.e., N (Q) = X ), so that the natural quotient map π : X → X /N (Q) is a quotient transformation by (f). Thus X

Q

−−−→  π 

Y X /N (Q) .

Therefore according to Section 5.1 and using (g) we get Y∼ = X /N (Q) ⇐⇒ Q ∼ = π =⇒ Q is a quotient map. (i) Conversely, suppose Q ∈ B[X , Y] is a quotient transformation (and hence R(Q) = Y is trivially closed) between Banach spaces. By Theorem 5.10(b) and its proof we get: (1) Y ∼ = X /N (Q) where ∼ = means topologically isomorphic,  and (2) Q : X /N (Q) → Y is the topological isomorphism of X /N (Q) onto Y  ◦ π and Q  = Q. In this case Q = 1 by (e2 ), and so such that Q = Q  = 1. This ensures that Q  is an isometry, as shown in Claim 2 below. In Q   fact, note that Q[x] = Q(π(x)) = Qx for every [x] ∈ X /N (Q) as π is surjective. Claim 1. If x ∈ Q−1 ({Qx}) for some x ∈ X , then [x ] = [x].

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 = Qx Proof of Claim 1 . Suppose x ∈ Q−1 ({Qx}) for some x ∈ X . Since Q[x]  −1      for every x ∈ X , then x ∈ Q ({Q[x]}) so that Q[x ] = Qx ∈ {Q[x]} (as Q is   is invertible).   ] = Q[x], which implies [x ] = [x] (as Q   surjective). Hence Q[x    Claim 2. Q[x] = [x]  for every [x] ∈ X /N (Q). Proof of Claim 2 . Take x ∈ X and ε > 0 arbitrary. By (c2 ), there is a vector xε ∈ Q−1 ({Qx}) ⊆ X such that Qx = Qxε and xε  < Qx + ε. So by Claim 1           [x]  + ε = [x]  + ε [x]  = [xε ]  = xε  < Qx + ε = Q[x] + ε ≤ Q    = 1. Hence −ε < Q[x]  for every [x] ∈ X /N (Q) because Q − [x]  < 0 < ε     − [x]   < ε. Since ε > 0 is arbitrary (it does not depend on and so  Q[x]    [x]), we get Q[x] = [x]  for every [x] ∈ X /N (Q).    : X /N (Q) → Y is an isometry, thus an Then the topological isomorphism Q isometric isomorphism. So X /N (Q) and Y are isometrically isomorphic. (j) Consider the diagram in the proof of Theorem 5.1 and the isometric isomor Let B  be an arbitrary open unit ball in phisms J : X → X and K : Y → Y. X     X . Since X ⊆ X , BX = BX ∩ X is an open unit ball in X. So BX = J −1 (BX ) is an open unit ball in X . If Q ∈ B[X , Y] is a quotient transformation, then BY = Q(BX ) is an open unit ball in Y by (d). Again, BY = K(BY ) is an open  and therefore K(BY ) = B  ∩ Y for some open unit ball B  in unit ball in Y, Y Y    Hence Y because Y ⊆ Y.   ∩ X). B  ∩ Y = K(BY ) = K(Q(BX )) = K(Q(J −1 (B  ∩ X))) = Q(B Y

X

X

  ). As B  is an arbitrary open unit So by continuity and denseness, BY = Q(B X X     ball in X , it follows that Q ∈ B[X , Y] is a quotient transformation by (d).   Remark 5.14. Quotient Topology. Let TZ denote the topology of a topological space Z. A topological quotient map is a function Q : X → Y of a topological space X onto a topological space Y such that B ∈ TY ⇐⇒ Q−1 (B) ∈ TX (i.e., a subset B of Y is open in Y if and only if its inverse image under Q is open in X). If X and Y are topological spaces and if there is a topological quotient map Q of X onto Y, then the topology of Y is called the quotient topology induced by Q. Thus if a surjective map Q : X → Y (i.e., Q(Q−1 (B)) = B for every B ⊆ Y ) is a continuous (i.e., B ∈ TY =⇒ Q−1 (B) ∈ TX ) open map (i.e., A ∈ TX =⇒ Q(A) ∈ TY ) of a topological space X onto a topological space Y, then Q is a topological quotient map. This definition does not involve the norm condition of Definition 5.12, and so imposes no restriction to norms (since norms play no role in general topological spaces). But a quotient transformation of a normed space X onto a normed space Y is a topological quotient map by Theorem 5.13; in particular, if M is a proper subspace of X , then the natural quotient map π : X → X /M is a topological quotient map and the topology of X /M is the quotient topology induced by π.

5.4 Additional Propositions

101

5.4 Additional Propositions It is a (true) clich´e to say that the three fundamental theorems in functional analysis are the Open Mapping Theorem, the Banach–Steinhaus Theorem and the Hahn–Banach Theorem. (These are all classical and really fundamental results found in essentially all books on functional analysis). Those aspects of the Hahn–Banach Theorem required in this book have been discussed in Remark 5.2 (and will be discussed again in the next chapter). The other two (the Open Mapping and the Banach–Steinhaus Theorems, that is) will be stated in this section together with some of their immediate consequences (only those that are required in the sequel). Recall: a mapping between topological spaces is continuous if the inverse image of every open set is open, a mapping between topological spaces is open if the image of every open set is open. Proposition 5.A. (Open Mapping Theorem). If X and Y are Banach spaces and T ∈ B[X , Y] is surjective, then T is an open mapping. A crucial corollary of the Open Mapping Theorem says that between Banach spaces, an invertible bounded linear transformation has a bounded inverse. Proposition 5.B. (Inverse Mapping Theorem). If X and Y are Banach spaces and T ∈ B[X , Y] is injective and surjective, then T −1 ∈ B[Y, X ]. This is partially responsible for the next result, which plays a major role in establishing the notion of spectrum of an operator. Recall: a linear transformation T of a normed space X into a normed space Y is bounded if there is a constant β > 0 such that T x ≤ βx for every x ∈ X , and it is bounded below if there is a constant α > 0 such that αx ≤ T x for every x ∈ X . Proposition 5.C. (Bounded Inverse Theorem). Let X and Y be Banach spaces and take any T ∈ B[X , Y]. The following assertions are equivalent. (a) T has a bounded inverse on its range. (b) T is bounded below . (c) T is injective and has a closed range. A further application of the Open Mapping Theorem that will be required later is the Closed Graph Theorem. The graph of a linear transformation L between linear spaces X and Y is the subset of the Cartesian product X ×Y

  GL = (x, y) ∈ X ×Y : y = Lx = (x, Lx) ∈ X ×Y : for every x ∈ X . The set GL is said to be closed if it is closed in the metric space (X ×Y, d), or in the normed space (X ⊕Y,  · ), which boils down to the same thing (see Section 5.1). There is, however, a significant difference: the set GL is a linear manifold of the normed space (X ⊕Y,  · ) by linearity of L. If X and Y are Banach spaces, then their direct sum X ⊕Y equipped with any of the standard norms

102

5. Normed-Space Results

 ·  of Section 5.1 is again a Banach space (see, e.g., [52, Example 4.E]). So if the graph GL of a linear transformation L is regarded as a subset of the direct sum X ⊕Y of Banach spaces X and Y equipped with any of those standard norms, and if GL is closed, then GL is a Banach space. Proposition 5.D. (Closed Graph Theorem). If X and Y are Banach spaces and T : X → Y is linear, then T is continuous if and only if GT is closed. Another useful inverse theorem establishes a power series expansion for (λI − T )−1 if T  < |λ| (this condition will be weakened in Proposition 5.P). Proposition 5.E. (Neumann Expansion). If T ∈ B[X ] is an operator on a Banach space X and λ is any scalar such that T  < |λ|, then λI − T has an inverse in B[X ] given by the following uniformly convergent series:  ∞  k T . (λI − T )−1 = λ1 λ k=0

Proposition 5.F. (Banach–Steinhaus Theorem). Let {Tω }ω∈Ω be an indexed family (indexed by an arbitrary index set Ω) of bounded linear transformations of a Banach space X into a normed space Y. If supω∈Ω Tω x < ∞ for every x ∈ X , then supω∈Ω Tω  < ∞. A sequence of linear transformations of a normed space X into a normed space Y (i.e., an L[X , Y]-valued sequence) {Ln } is pointwise convergent if for every x ∈ X the Y-valued sequence {Ln x} converges in Y to, say, yx ∈ Y. Since the limit is unique, this defines a transformation L : X → Y given by L(x) = yx for every x ∈ X . It is readily verified that L is linear. Therefore an L[X , Y]-valued sequence {Ln } is pointwise convergent if and only if there exists L ∈ L[X , Y] such that (Ln − L)x → 0 for every x ∈ X . Now consider a sequence {Tn } of bounded linear transformations of a normed space X into a normed space Y (i.e., a B[X , Y]-valued sequence). An important application of the Banach–Steinhaus Theorem ensures that if X is a Banach space and {Tn } is pointwise convergent, then the limit T is bounded. Proposition 5.G. If X is a Banach space and Y is a normed space, then a B[X , Y]-valued sequence {Tn } is pointwise convergent if and only if there is a T ∈ B[X , Y] such that (Tn − T )x → 0 for every x ∈ X . Let X and Y be a normed spaces. If a B[X , Y]-valued sequence {Tn } is such that there exists T ∈ B[X , Y] for which (Tn − T )x → 0 for every x ∈ X , then the sequence {Tn } is strongly convergent, or it is said to converge strongly to T s T ). If {Tn } converges to T in the induced uniform norm on (notation: Tn −→ B[X , Y] (i.e., if Tn − T  → 0), then {Tn } is uniformly convergent, or it is said u T ). It is clear that to converge uniformly to T ∈ B[X , Y] (notation: Tn −→ u Tn −→ T

=⇒

s Tn −→ T,

and the converse fails. If {Tn } is strongly convergent, then it is strongly bounded (i.e., (Tn − T )x → 0 implies Tn x → T x, which in turn implies

5.4 Additional Propositions

103

supn Tn x < ∞ for every x ∈ X ). If X is a Banach space, then the Banach– Steinhaus Theorem ensures that strong boundedness implies uniform boundedness (i.e., supn Tn x < ∞ for every x ∈ X implies supn Tn  < ∞, and the converse holds trivially even for a plain normed space X ). Therefore if X is a Banach space, then the Banach–Steinhaus Theorem ensures that pointwise convergence coincides with strong convergence (Proposition 5.G), and strong boundedness coincides with uniform boundedness (Proposition 5.F). Thus s T Tn −→

=⇒

supn Tn  < ∞ whenever X is Banach.

An important special case of the above notions refers to power sequences of operators. An operator T ∈ B[X ] on a normed space X is uniformly stable or strongly stable if the power sequence {T n } converges uniformly or strongly to the null operator, and it is power bounded if the power sequence is bounded. u s O, T n −→ O, and supn T n  < ∞, respectively. In other words, if T n −→ With X and Y being normed spaces, let B0 [X ,Y] stand for the collection of finite-rank bounded linear transformations and B∞ [X ,Y] for the collection of compact linear transformations (for the definition of compact linear transformation, see, e.g., [52, Section 4.9]). Recall: B0 [X ,Y] ⊆ B∞ [X ,Y] ⊆ B[X ,Y]. As before, set B0 [X ] = B0 [X , X ], B∞ [X ] = B∞ [X , X ], and B[X ] = B[X , X ]. Proposition 5.H. Let X be a normed space and Y be a Banach space. If a sequence of finite-rank bounded linear transformations converges uniformly, then u T with Tn ∈ B0 [X , Y], then T ∈ B∞ [X , Y]). the limit is compact (i.e., if Tn −→ A metric space is separable if it has a countable dense subset. Proposition 5.I. The range of a compact linear transformation is separable. A sequence {xk } of vectors in a normed space X is a Schauder basis for X if for n sequence of scalars {αk (x)} such that x =  each x in X there is a unique α (x)x (i.e., x = lim k n k k k=1 αk (x)xk ). Every Schauder basis for X is a sequence of linearly independent vectors in X that spans X (i.e., (span {xk })− = X ). So if a normed space has a Schauder basis, then it is separable. The converse of Proposition 5.H holds if the Banach space Y has a Schauder basis. Proposition 5.J. If Y is a Banach space with a Schauder basis and X is any normed space, then every compact transformation T ∈ B∞ [X , Y] is the uniform limit of a sequence of finite-rank transformations Tn ∈ B0 [X , Y]. Does every separable (infinite-dimensional) Banach space have a Schauder basis? This is a famous question, raised by Banach himself in the early 1930s, that remained open for a long period. Each separable Banach space that ever came up in analysis during that period (and this includes all classical examples) had a Schauder basis. The surprising negative answer to that question was given by Enflo [25] in 1973, who constructed a separable Banach space that has no Schauder basis. In fact, he exhibited a separable (and reflexive) Banach space X for which B0 [X ] is not dense in B∞ [X ], so that there exist compact

104

5. Normed-Space Results

operators on X that are not the (uniform) limit of finite-rank operators. Hence the converse of Proposition 5.H fails in general, and by Proposition 5.J such an X is a separable Banach space without a Schauder basis. Proposition 5.J induces a pair of central notions. A bounded linear transformation is approximable if it is the uniform limit of a sequence of finite-rank bounded linear transformations. A Banach space Y has the approximation property if every compact linear transformation of an arbitrary normed space X into Y is approximable. In other words, a Banach space Y has the approximation property if every T ∈ B∞ [X , Y] for every normed space X is such that u T . Equivalently, a there is a B0 [X , Y]-valued sequence {Tn } for which Tn −→ Banach space Y has the approximation property if B0[X,Y] is dense in B∞[X,Y]. By Proposition 5.J (and Proposition 5.I) we may infer the following results: (i) Banach spaces with a Schauder basis have the approximation property. (ii) Hilbert spaces have the approximation property. We explore the approximation property further in Remark 8.14 (Chapter 8). From now on up to Proposition 5.P, X is a complex Banach space and T is an operator in the unital complex Banach algebra B[X ] (of bounded linear transformations of X into itself). Let I stand for the identity operator in B[X ]. The resolvent set ρ(T ) of an operator T in B[X ] is the set of all scalars λ in C for which the operator λI − T in B[X ] is invertible (i.e., has a bounded inverse on X according to the Inverse Mapping Theorem):

 ρ(T ) = λ ∈ C : λI − T has an inverse in B[X ] 

= λ ∈ C : N (λI − T ) = {0} and R(λI − T ) = X . The spectrum σ(T ) of T ∈ B[X ] is the complement of the resolvent set, σ(T ) = C \ρ(T )

 = λ ∈ C : λI − T has no inverse in B[X ] 

= λ ∈ C : N (λI − T ) = {0} or R(λI − T ) = X . Proposition 5.K. ρ(T ) is nonempty and open, and σ(T ) is compact (i.e., closed and bounded in C ). If X is nonzero, then σ(T ) is nonempty. A complex number λ for which λI − T is not injective (i.e., for which λI − T has no left inverse or, equivalently, for which N (λI − T ) = {0}) is an eigenvalue of T . The set of all eigenvalues of T is the point spectrum of T ,

 σP (T ) = λ ∈ C : N (λI − T ) = {0} . If λI − T is injective, then λI − T has an inverse on its range which may or not be bounded. By the Bounded Inverse Theorem, it is bounded if and only if R(λI − T ) is closed. If it is not bounded but has a dense range, then set

5.4 Additional Propositions

105

 σC (T ) = λ ∈ C : N (λI − T ) = {0}, R(λI − T )− = X and R(λI − T ) = X , which is the continuous spectrum of T . The residual spectrum of T is the set σR (T ) = σ(T )\(σP (T ) ∪ σC (T ))

 = λ ∈ C : N (λI − T ) = {0} and R(λI − T )− = X . It is clear that {σP (T ), σC (T ), σR (T )} forms a partition of σ(T ), referred to as the classical partition of the spectrum. If Λ is a subset of C , and p : C → C is a polynomial with complex coefficients, then set p(Λ) = {p(λ) ∈ C : λ ∈ Λ}. Proposition 5.L. (Spectral Mapping Theorem for Polynomials). If p is an arbitrary polynomial with complex coefficients, then σ(p(T )) = p(σ(T )). The spectral radius of an operator T in B[X ] is the nonnegative number r(T ) = supλ∈σ(T ) |λ|. An application of the Spectral Mapping Theorem for Polynomials and the Neumann expansion yields the following expressions. Proposition 5.M. For every nonnegative integer n r(T )n = r(T n ) ≤ T n  ≤ T n

and so

supn T n  < ∞ =⇒ r(T ) ≤ 1. 1

Proposition 5.N. (Gelfand–Beurling Formula). {T n  n} converges and 1

r(T ) = limn T n  n . The Gelfand–Beurling Formula fully characterizes uniform stability. Proposition 5.O. The following assertions are pairwise equivalent. u (a) T n −→ O.

(b) r(T ) < 1. (c) T n  ≤ βαn for every n ≥ 0, for some β ≥ 1 and some α ∈ (0, 1). 5.P below extends the Neumann expansion. The infinite series ∞Proposition T k ( ) is said to converge uniformly or strongly if the sequence of partial k=0  λ  n T k ∞ sums ( ) n=0 converges uniformly or strongly, respectively. In either k=0 λ  ∞ case we denote by k=0 ( Tλ )k the (uniform or strong) limit operator in B[X ]. Proposition 5.P. Take any operator T and any nonzero scalar λ. ∞ T k (a) r(T ) < |λ| if and only if ) converges uniformly. In this case, λ k=0 ( λ ∞ lies in ρ(T ) and (λI − T )−1 = λ1 k=0 ( Tλ )k . ∞ T k ( ) converges strongly, then λ lies in ρ(T ) and (b) If r(T ) = |λ| and ∞ k=0 λ (λI − T )−1 = λ1 k=0 ( Tλ )k . ∞ (c) If |λ| < r(T ), then k=0 ( Tλ )k does not converge strongly.

106

5. Normed-Space Results

As in Section 2.4, an inner product on a linear space X is a Hermitian symmetric sesquilinear form · ; · : X ×X → F generating a positive quadratic form  ·  : X → R given by x = x ; x 1/2 for every x ∈ X , which is a norm on X (and the inner product is a continuous sesquilinear form by the Schwartz inequality: | x ; y | ≤ xy for every x, y ∈ X ). The pair (X , · ; · ) is an inner product’ space — which is a special case of a normed space (X ,  · ). A Hilbert space is a complete inner product space, which is complete with respect to the norm generated by the inner product. Thus a Hilbert space is a Banach space whose norm is induced by an inner product. If F = C or F = R , then we refer to a complex or real Hilbert space. Two vectors x, y in an inner product space X are orthogonal (notation: x ⊥ y) if their inner product x ; y is zero. If M is a linear manifold of an inner product space X , then its orthogonal complement M⊥ is the (closed) subspace made up of all vectors that are orthogonal to every vector in M; that is, M⊥ = {v ∈ X : v ; u = 0 for every u ∈ M}. The next proposition is the central result of Hilbert-space geometry. Proposition 5.Q. (Projection Theorem). If M is a subspace of a Hilbert space X , then X = M + M⊥ . Since M and M⊥ are trivially algebraically disjoint (i.e., M ∩ M⊥ = {0}), then Proposition 5.Q ensures that the (closed) subspace M⊥ is a complement (an orthogonal complement) of an arbitrary (closed) subspace M of X , so that every subspace of a Hilbert space is complemented, and therefore all Hilbert spaces are complemented (and, as we saw Remark 5.3(a), only they are). Since M and M⊥ are algebraic complements, their ordinary sum is isomorphic to their direct sum, and the natural isomorphism between them is, in fact, an isometric isomorphism — where the norm of x = u + v with u in M and v in M⊥ is such that x2 = u2 + v2 (by the Pythagorean Theorem). Also, the inner product in a direct sum of orthogonal linear manifolds, say in M ⊕ M⊥, is given by (u1 , v1 ) ; (u2 , v2 ) = u1 ; u2 + v1 ; v2 for every u1 , u2 in M and v1 , v2 in M⊥. An isometric isomorphism between inner product spaces preserves inner product, is called a unitary transformation, and isometrically isomorphic inner product spaces are called unitarily equivalent. Proposition 5.Q can be rewritten (with ∼ = replaced by = as we did before) in the form X = M ⊕ M⊥ . (Here ∼ = , which has been replaced by = , means isometrically isomorphic, i.e., unitarily equivalent.) If M and N are orthogonal subspaces (or orthogonal linear manifolds) of an inner product space (i.e., if u ; v = 0 for every u ∈ M and every v ∈ N — notation: M ⊥ N ), then M ⊕ N is referred to as the orthogonal direct sum of M and N (which is an internal direct sum). Since M ⊥ M⊥ tautologically, the above decomposition is referred to as orthogonal decomposition, and this leads to the notation M⊥ = X  M whenever M is a subspace of a Hilbert space X . An orthogonal projection E on an inner product space X is a projection for which R(E) ⊥ N (E).

5.4 Additional Propositions

107

Proposition 5.R. (i) If E ∈ L[X ] is a nonzero orthogonal projection on an inner product space X , then E ∈ B[X ] and E = 1. Moreover, (ii) if E ∈ B[X ] is a nonzero projection on a Hilbert space X , then E is orthogonal if and only if E = 1. Furthermore, (iii) if M is a subspace of a Hilbert space X , then there is a unique orthogonal projection E ∈ B[X ] with R(E) = M. If M is a subspace of a Banach space X , then a minimal projection E onto M (i.e., with R(E) = M) is a continuous projection E ∈ B[X ] such that E ≤ P  for every projection P ∈ B[X ] with R(P ) = M. A subspace M is 1-complemented if there is a projection E ∈ B[X ] with R(E) = M and E = 1 (which is minimal since P  ≥ 1 for every projection P ∈ B[X ]). So part (b) of Theorem 5.4 says: if M− is 1-complemented, then T = T . Proposition 5.S. In an arbitrary Banach space, minimal projections on complemented subspaces do not necessarily exist, are not necessarily unique if they exist, and there are minimal projections of norm greater than one. In a Hilbert space, however, Proposition 5.R says that minimal projections exist, are orthogonal, unique, and have norm 1. A set of mutually orthogonal vectors is an orthogonal set. It is an orthonormal set if all vectors in it have norm one. Every Hilbert space has an orthonormal basis (i.e., an orthonormal set whose span is dense in the space). A Hilbert space is separable if and only if it has a countable orthonormal basis. Proposition 5.T. (The Fourier Series Theorem). Let {eγ }γ∈Γ be an orthonormal set in a Hilbert space X . The assertions below are pairwise equivalent. (a) {eγ }γ∈Γ is an orthonormal basis for X .  (b) x = γ∈Γ x ; eγ eγ for every x ∈ X , referred to as the Fourier series expansion of x, which is unique in terms of {eγ }γ∈Γ , and the scalars in { x ; eγ }γ∈Γ are the Fourier coefficients of x with respect to {eγ }γ∈Γ .  (c) x ; y = γ∈Γ x ; eγ y ; eγ for every pair of vectors x, y in X .  (d) x2 = γ∈Γ | x ; eγ |2 for every x ∈ X . This is the Parseval identity. (e) Every linear manifold of X that includes {eγ }γ∈Γ is dense in X . It is readily verified that a bounded linear functional f on an inner product space X (i.e., an f ∈ X ∗ ) is naturally associated with each vector y ∈ X and f  = y. The next result is central in the Hilbert-space theory and asserts that if X is complete (i.e., if X is a Hilbert space), then the converse holds. Proposition 5.U. (The Riesz Representation Theorem). If X is a Hilbert space, then for every f in the dual X ∗ of X there is a unique y in X such that f (x) = x ; y

for every

x ∈ X.

Moreover, f  = y. Such a unique vector y in X is called the Riesz representation of the functional f in X ∗ = B[X , F ].

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5. Normed-Space Results

The result below is a major corollary of the Riesz Representation Theorem. Proposition 5.V. For every Hilbert space X there exists a surjective (not necessarily linear) isometry Ψ : X ∗ → X of X ∗ onto X , which is additive and conjugate homogeneous (i.e., Ψ (α f ) = aΨ (f ) for every f ∈ X ∗ and α ∈ F ). The Riesz Representation Theorem allows us to redefine a notion of adjoint for bounded linear transformations of a Hilbert space into an inner product space. If X is a Hilbert space and Y is an inner product space, then the Hilbertspace adjoint of T ∈ B[X , Y] is the (unique) operator T ∗ ∈ B[Y, X ] such that T x ; y = x ; T ∗ y for every x ∈ X and y ∈ Y — the left-hand side inner product equips the inner product space Y and the right-hand side the Hilbert space X . Basic properties of the adjoint, when Y is also a Hilbert space, are: T ∗∗ = T

and

T ∗ 2 = T ∗ T  = T T ∗  = T 2 .

Remark 5.W. On the Notions of Normed-Space and Hilbert-Space Adjoints. (a) The Riesz Representation Theorem and its corollary in Proposition 5.V ensure that every Hilbert space is isometrically equivalent to its dual. (In particular, every real Hilbert space is isometrically isomorphic — i.e., unitarily equivalent — to its dual). The Riesz Representation Theorem also ensures that if X is a Hilbert space, then for each T ∈ B[X , Y] into any inner product space Y there exists a unique T ∗ ∈ B[Y, X ] such that T x ; y = x ; T ∗ y for every x ∈ X and y ∈ Y, as above. This T ∗ ∈ B[Y, X ] is the Hilbert-space adjoint of T ∈ B[X , Y]. In particular, for Y = X such an inner product identity defines the Hilbert-space adjoint T ∗ ∈ B[X ] of every operator T ∈ B[X ] acting on a Hilbert space X . But the Riesz Representation Theorem works only if X is a Hilbert space; not if X is an inner product space. Outcome: there is no Hilbert-space adjoint for operators acting on inner product spaces. (b) Let X and Y be normed spaces. The normed-space adjoint T ∗ ∈ B[Y ∗, X ∗ ] of T ∈ B[X , Y] was defined in Section 5.1. It is the restriction of the algebraic adjoint T  ∈ L[Y  , X  ] (cf. Section 1.1) to Y ∗ ⊆ Y  , which is the unique transformation in B[Y ∗, X ∗ ] for which T ∗g = g T for every g ∈ Y ∗ (i.e., for which (T ∗g)(x) = g(T x) — also written as x ; T ∗ g = T x ; g which, of course, does not mean inner product — for every g ∈ Y ∗ and every x ∈ X ). See, e.g., [70, Definition 3.13]. In particular, for Y = X the normed-space adjoint of an operator T ∈ B[X ] on a normed space X is the operator T ∗ ∈ B[X ∗ ] on the Banach space X ∗. Such a definition holds tautologically if X is a Banach space. If X is a Hilbert space, then we may compare the normed-space adjoint with the Hilbert-space adjoint (which makes sense only if X is a Hilbert space). (c) This is a major difference between normed-space and Hilbert-space ad joints: normed-space adjoints are defined for transformations T ∈ B[X , Y] acting on any normed space X , while Hilbert-space adjoints are defined only for transformations acting on a Hilbert space X . (For the special case of Y = X , such a difference is linked to the fact that the normed-space adjoint T ∗ ∈ B[X ∗ ]

5.4 Additional Propositions

109

always acts on a Banach space X ∗, while the Hilbert-space adjoint T ∗ ∈ B[X ] acts on the same Hilbert space X upon which T acts). A minor difference is that for the normed-space adjoint (αT )∗ = αT ∗, while for the Hilbert-space adjoint (αT )∗ = α T ∗, for every α ∈ F. (By the way, the other basic properties listed in Section 5.1 coincide for both notions, viz., T ∗  = T , (T + S)∗ = T ∗ + S ∗, (T S)∗ = S ∗ T ∗, and I ∗ is the identity operator if Y = X .) (d) A special instance of such a major difference is this. If T ∈ B[X ] acts on any normed space (in particular, on an inner product space), then its normed-space ∗ ∗ adjoint is such that T∗ = T . (Actually, T ∼ = T ∗ = T∗ according to Theorem ∗ 5.1(f) and, since T acts on the Banach space X ∗, the identity T ∗ = T∗ is trivial — which holds even for the Hilbert-space adjoint when T ∗ acts on a Hilbert space X .) Now suppose T acts on an incomplete inner product space X . We can extend T to get T on the Hilbert space X, and then take its Hilbert-space ∗ adjoint T . But we cannot take the Hilbert-space adjoint of T (because it does not act on a Hilbert space), and hence the expression T∗ does not make sense ∗ in this case. Consequently, the identity T∗ = T only makes nontrivial sense ∗ (i.e., for X not complete) when T stands for the normed-space adjoint. For an arbitrary subset Λ of the complex plane C set Λ∗ = {λ ∈ C : λ ∈ Λ}, the set of all complex conjugates of elements from Λ. Proposition 5.X. If T ∈ B[X ] where X is a complex Hilbert space, then σ(T ) = σ(T ∗ )∗ ,

σC (T ) = σC (T ∗ )∗ ,

σR (T ) = σP (T ∗ )∗ \σP (T ).

Remark 5.Y. On Point and Residual Spectra. (a) The point spectrum of a linear transformation L ∈ L[X ] acting on a linear space X is the set of its eigenvalues, viz., σP (L) = {λ ∈ F : N (λI − L) = {0}}. If T ∈ B[X ] is a bounded linear operator acting on a normed space X , then its point spectrum is identically defined, even if X is not a Banach space, σP (T ) = {λ ∈ F : N (λI − T ) = {0}}. If X is the completion of a normed space X and T ∈ B[X] is the extension over completion of T ∈ B[X ], then σP (T) = {λ ∈ F : N (λI − T) = {0}},  = λI on X, and so write where we use the same notation λI for its extension λI     λI − T = λI − T = λI − T . Recall from Theorem 5.1(i) that N (λI − T ) ⊆ N (λI − T) (and the reverse inclusion of closures holds if R(λI − T ) is closed). Thus N (λI − T ) = 0 implies N (λI − T) = 0. Hence σP (T ) ⊆ σP (T). (For further results on the change of the spectrum under extensions in general — not necessarily extensions over completion — see [37, Theorem 2.16.2].)

110

5. Normed-Space Results

(b) However, we can only consider the spectrum σ(T) of T, since spectra are defined in a unital complex Banach algebra, thus defined only for operators acting on a Banach space. (Although point spectra are defined for operators on normed spaces, spectra are not.) If T ∈ B[X ] acts on a Hilbert space X , then we get the residual spectrum formula in Proposition 5.X, σR (T ) = σP (T ∗ )∗ \σP (T ), with respect to the Hilbert-space adjoint T ∗ ∈ B[X ]. Since this expression for the residual spectrum is given entirely in terms of point spectra, and since point spectra are well-defined for operators on linear spaces, it might be tempting to try a definition of the residual spectrum on inner product spaces (not necessarily complete), just by mirroring the above residual spectrum formula which is based on point spectra only. The trouble, however, comes from another side. Hilbert-space adjoints T ∗ ∈ B[X ] are defined only on a Hilbert space, not on an inner product space (see Remark 5.W). Outcome: there is no generalization of the residual spectrum formula for inner product spaces. We close this section with a simple result that will be required in Chapter 9. Let X and Y be linear spaces and equip each of them with a pair of norms, say,  ·  X1 and  ·  X2 on X and  · Y1 and  · Y2 on Y. Take a nonzero linear transformation T ∈ L[X , Y] and suppose it is bounded when the linear spaces X , Y are both equipped with the 1-norms or both equipped with the 2-norms (i.e., T ∈ B[(X ,  · X1 ), (Y,  · Y1 )] and T ∈ B[(X ,  · X2 ), (Y,  · Y2 )]). Let  · 1 and  · 2 be the induced uniform norms in B[(X ,  · X1 ), (Y,  · Y1 )] and B[(X ,  · X2 ), (Y,  · Y2 )], respectively. Proposition 5.Z. If  · X2 ≤  · X1 and  · Y2 ≤  · Y1 , T 1 ≤ sup

x ∈N (T )

T xY1 T xY2

T 2

and

then

x

T 2 ≤ sup xXX1 T 1 ; x=0

2

equivalently, T xY2 inf x ∈N (T ) T xY1

x

T 1 ≤ T 2 ≤ sup xXX1 T 1 . x=0

2

Notes: Section 5.4 has been focused on bounded linear transformations only, taking up mainly the topics of Section 5.2, with some fundamental results in mind that are required in this and in subsequent chapters. Since all propositions listed in this section (plus the Hahn–Banach Theorem in Remark 5.2(d)) are classic, well-known, and indeed fundamental statements, they can be found in a variety of texts. The books listed below represent just a few samples among a huge collection of other possible alternatives. For the classical results in Propositions 5.A to 5.J see, e.g., [52, Theorems 4.21, 4.22, Corollary 4.24, Theorem 4.25, Problem 4.47, Theorems 4.43, 4.44, Corollary 4.55, Proposition 4.57, and Problem 4.58], respectively. These (together with Remark 5.2) comprise an introductory basic toolbox on Banachspace operator theory, being mandatory in every functional analysis course.

5.4 Additional Propositions

111

Propositions 5.A to 5.E are related to bounded inverses of bounded linear transformations, while Propositions 5.F and 5.G refer mainly to sequences of bounded linear transformations acting on Banach spaces. Proposition 5.A is the Open Mapping Theorem, whose proof comes from the Baire Category Theorem on metric spaces (see, e.g., [52, Theorem 3.58]), and the inverse theorems in Propositions 5.B and 5.C (the Inverse Mapping Theorem and the Bounded Inverse Theorem) are consequences of it, as is the Closed Graph Theorem in Proposition 5.D. The Neumann Expansion (due to C.G. Neumann) in Proposition 5.E has a proof based on another basic result which says that a normed space is complete if and only if every absolutely summable sequence is summable (see, e.g., [52, Proposition 4.4]). The Baire Category Theorem also yields a classical proof for the Banach–Steinhaus Theorem of Proposition 5.F, which in turn leads to the notion of strong convergence in Proposition 5.G. The approximation results in Propositions 5.H and 5.J use the fact that if Y is a Banach space, then B∞ [X , Y] is a (closed) subspace of B[X , Y]. Recall that if T is an operator on a normed space X , and if p : F → F is a polynomial with n coefficients αi in F , then the associated polynomial of T , viz., p(T ) = i=0 αi T i (as in Proposition 5.L) is again an operator on X . Propositions 5.K to 5.P are the very first steps towards a spectral theory of operators on a complex Banach space. See, e.g., [56, Theorems 2.1, 2.2, Theorem 2.7, Corollary 2.9, Theorem 2.10, Corollaries 2.11 and 2.12]. The role played by Proposition 5.C in characterizing the spectrum of bounded linear operators on a complex Banach space is highlighted by the classical partition of the spectrum into point, continuous, and residual spectra. The Hilbert-space geometry is centered around the Projection Theorem of Proposition 5.Q, which can be equivalently stated as in part (iii) of Proposition 5.R. This justifies the name Projection Theorem and distinguishes Hilbert spaces as the prototype of complemented Banach spaces. For Propositions 5.Q and 5.R see, e.g., [52, Theorems 5.20, 5.52 and Propositions 5.51, 5.81]. Proposition 5.S shows that such a nice scenario vanishes in an arbitrary Banach space (see, e.g., [72, Introduction, §1]). The section closes with a full version of the Fourier Series Theorem in Proposition 5.T, with the Riesz Representation Theorem in Hilbert space in Proposition 5.U and a corollary of it in Proposition 5.V, and with three fundamental results on parts of the spectrum for Hilbert-space adjoints in Proposition 5.X (see, e.g., [52, Theorems 5.48, 5.62, 5.63, 6.14]). These will often be required in Sections 7.4, 9.2 and 9.3. Proof of Proposition 5.Z. Since  · X2 ≤  · X1 and T = O, T 1 = sup

T xY1 xX1

≤ sup

T xY2 xX2

x=0 x=0

equivalently,

≤ sup x=0

T xY1 xX2

sup x ∈N (T )

T xY1 T xY2

=

sup x ∈N (T )

= T 2

T xY1 T xY2 xX2 T xY2

sup x ∈N (T )

T xY1 T xY2

;

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5. Normed-Space Results

T xY2 inf x ∈N (T ) T xY1

T 1 =

1

inf

T xY 1 T xY 2

x ∈N (T )

T 1 =

1 supx ∈N (T )

T xY 1 T xY 2

T 1 ≤ T 2 .

Moreover, since  · Y2 ≤  · Y1 and T xY1 ≤ T 1 xX1 for every x ∈ X , T 2 = sup x=0

T xY2 xX2

≤ sup x=0

T xY1 xX2

x

≤ T 1 sup xXX1 . x=0

2

 

p A final note. We use the notation p+ , or  (N ), or p (N 0 ) with p ≥ 1 for psummable scalar-valued sequences {ξk } (i.e., k |ξk |p < ∞). And the notation ∞ ∞ ∞ + , or  (N ), or  (N 0 ) for bounded sequences (i.e., supk |ξk | < ∞). We drop the subscript + when dealing with p-summable or bounded nets indexed by Z , rather than sequences indexed by N or N 0 , so that we use p for p (Z ) and ∞ for ∞ (Z ) (where N , N 0 and Z stand for the sets of positive integers, nonnegative integers, and all integers, respectively).

Suggested Readings Bachman and Narici [2] Brown and Pearcy [5] Conway [12], [13], [14] Dunford and Schwartz [23] Halmos [31] Harte [34] Kato [46] Kubrusly [52], [56]

Megginson [70] Reed and Simon [80] Rudin [82] Schechter [87] Simon [89] Taylor and Lay [91] Weidmann [92] Yosida [95]

6 Bounded Bilinear Maps

This chapter considers basic normed-space aspects of bilinear maps. The first notion is that of a bounded bilinear map, which is shown to coincide with the notion of continuous bilinear maps. The final targets are extension theorems for bounded bilinear maps with norm preservation. As in the previous chapters, all linear spaces (and therefore all normed spaces) are over the same field.

6.1 Boundedness and Continuity Let X , Y, Z be normed spaces and let b[X ×Y, Z] be the linear space of all bilinear maps φ from X ×Y to Z. A bounded bilinear map is a map φ in b[X ×Y, Z] for which there exists a positive constant β > 0 such that φ(x, y) ≤ βx y

for every

(x, y) ∈ X ×Y.

Equivalently, φ is bounded if sup

0=x∈X , 0=y∈Y

φ(x,y) x y

< ∞.

Let b[X ×Y, Z] denote the collection of all bounded bilinear maps from the linear space b[X ×Y, Z], which is a linear manifold of it, thus being itself a linear space. The next theorem sets forth the induced uniform norm on b[X ×Y, Z], thus making it into a normed space (which happens to be complete whenever Z is). As before, B[X , Z] denotes the normed space of all bounded linear transformations from the linear space L[X , Z]. Theorem 6.1. Let X , Y, Z be normed spaces (over the same field F ). (a) b[X ×Y, Z] is a linear space, the function  ·  : b[X ×Y, Z] → R given by   φ = inf β > 0 : φ(x, y) ≤ βx y for every (x, y) ∈ X ×Y , or equivalently, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Kubrusly, Bilinear Maps and Tensor Products in Operator Theory, Universitext, https://doi.org/10.1007/978-3-031-34093-2 6

113

114

6. Bounded Bilinear Maps

φ =

sup

0=x∈X , 0=y∈Y

φ(x,y) x y ,

for every φ ∈ b[X ×Y, Z] is a norm on b[X ×Y, Z], and so φ(x, y) ≤ φ x y

for every

(x, y) ∈ X ×Y.

(b) Moreover, b[X ×Y, Z] is a Banach space whenever Z is a Banach space. Proof. (a) Take φ, φ1 , φ2 ∈ b[X ×Y, Z] and α ∈ F arbitrarily. Clearly, αφ lies in b[X ×Y, Z]. As b[X ×Y, Z] is a linear space, φ1 + φ2 is bilinear. Consider the function  ·  : b[X ×Y, Z] → R defined by φ =

sup

0=x∈X , 0=y∈Y

φ(x,y) x y

+φ2 )(x,y) for every φ ∈ b[X ×Y, Z]. Since sup0=x∈X , 0=y∈Y (φ1x ≤φ1 +φ2 , y we get φ1 + φ2 lies in b[X ×Y, Z]. Hence b[X ×Y, Z] is a linear space. Now (i) φ ≥ 0, (ii) φ = 0 if and only if φ = 0, (iii) αφ = αφ, and (iv) φ1 + φ2  ≤ φ1  + φ2 . So  ·  is a norm on b[X ×Y, Z]. This is equivalently written as φ = inf{β > 0 : φ(x, y) ≤ βxy for every (x, y) ∈ X ×Y}, and therefore φ(x, y) ≤ φxy for every (x, y) ∈ X ×Y.

(b) Next take any b[X ×Y, Z]-valued Cauchy sequence {φn } and an arbitrary (x, y) in X ×Y. Thus φm (x, y) − φn (x, y) ≤ φm − φn  x y and hence {φn (x, y)} is a Cauchy sequence in Z, which converges in Z whenever Z is a Banach space. Then define a map φ : X ×Y → Z by φ(x, y) = limn φn (x, y) ∈ Z. This φ is bilinear. (Indeed, φ(x1 + x2 , y) = limn φn (x1 + x2 , y) = φ(x1 , y) + φ(x2 , y), and similarly φ(x, y1 + y2 ) = φ(x, y1 ) + φ(x, y2 ), and also φ(αx, y) = φ(x, αy) = αφ(x, y).) Moreover, this φ is bounded. In fact, take an arbitrary ε > 0. Since {φn } is a Cauchy sequence in b[X ×Y, Z], then (φm −φn )(x, y) ≤ φm − φn xy < εxy for every (x, y) ∈ X ×Y for m, n large enough. So (φm − φ)(x, y) = φm (x, y) − φ(x, y) = φm (x, y) − limn φn (x, y) = limn (φm − φn )(x, y) < εx y for m large enough. Thus φm − φ is bounded, and so is φ = φm − (φm − φ) (because φm ∈ b[X ×Y, Z]). Hence φ ∈ b[X ×Y, Z]. Furthermore, φm − φ =

sup

0=x∈X , 0=y∈Y

(φm − φ)(x,y) x y

0 there is a pair (xα , yα ) ∈ X ×Y such that αxα  yα  < φ(xα , yα ). In particular, for every positive integer k there exists a pair of nonzero vectors (xk , yk ) ∈ X ×Y such yk xk in X and yk = √ky in that 0 < xk  yk  < k1 φ(xk , yk ). Set xk = √kx k k   Y, and consider the sequences of nonzero vectors {xk } and {yk } which are such that xk → 0 in (X ,  · ) and yk → 0 in (Y,  · ). Equip the Cartesian product X ×Y with any of the equivalent product metrics d and consider the metric space (X ×Y, d), so that the sequence {(xk , yk )} is such that (xk , yk ) → (0, 0)

in

(X ×Y, d).

Since φ is bilinear, 0 < xk  yk  < k1 φ(xk , yk ) = xk  yk φ(xk , yk ) and hence 1 < φ(xk , yk ) for all positive integers k. Therefore

116

6. Bounded Bilinear Maps

φ(xk , yk ) → 0 = φ(0, 0)

in

(Z,  · ).

Hence φ : (X ×Y, d) → (Z,  · ) is not continuous at (0, 0). So (b) fails. To verify (c) =⇒(a), take an arbitrary X ×Y-valued convergent sequence {(xk , yk )}, say, (xk , yk ) → (x, y) in (X ×Y, d) for any equivalent product metric d. This means xk → x in (X ,  · ) and yk → y in (Y,  · ). Recall: convergent sequences are bounded. If the bilinear map φ is bounded, then φ(xk , yk ) − φ(x, y) ≤ φ(xk , yk ) − φ(xk , y) + φ(x, y) − φ(xk , y) = φ(xk , yk − y) + φ(x − xk , y)   ≤ φ (supk xk )yk − y + xk − x y , and so φ(xk , yk ) → φ(x, y) in (Z,  · ). Therefore φ : (X ×Y, d) → (Z,  · ) preserves convergence, which means φ is continuous. Finally we show that (c) =⇒ (d)

(d) with one of X or Y being Banach =⇒ (c).

and

The implication (c) =⇒ (d) is readily verified. Indeed, fix any 0 = y ∈ Y. So sup

0=x∈X

φy x x y

= sup

0=x∈X

φ(x,y) x y

≤

sup

0=x∈X , 0=y∈Y

φ(x,y) x y

= φ

whenever (c) holds. Thus the linear transformation φy : X → Z is bounded. Symmetrically, (c) implies φx ∈ B[Y, Z]. So (c) implies (d), and in this case φy  ≤ φy

and

φx  ≤ φx

for each x ∈ X and each y ∈ Y. Conversely, suppose (d) holds. Take an arbiy in Y, and consider the associated bounded trary nonzero y ∈ Y, set y  = y y linear transformation φ ∈ B[X , Z]. Thus 

φy x = φ(x, y  ) = φx y   ≤ φx  y   = φx 

for every

x ∈ X.



So the family {φy }0=y∈Y of transformations in B[X , Z] is pointwise bounded: 

sup φy x < ∞

for every

0=y∈Y

x ∈ X.

Suppose X is a Banach space to allow an application of the Banach–Steinhaus Theorem (cf. Proposition 5.F), which ensures uniform boundedness given pointwise boundedness for any family of bounded linear transformations of a Banach space into a normed space. Thus the above pointwise boundedness implies uniform boundedness by the Banach–Steinhaus Theorem: 

sup φy  < ∞.

0=y∈Y

Since

φ(x,y) x y



y

) = φ(x,y = φxx ≤ φy  for every (0, 0) = (x, y) ∈ X ×Y, x 

6.2 Identification with Bounded Linear Transformations

sup

0=x∈X , 0=y∈Y

φ(x,y) x y

117



≤ sup φy  < ∞, 0=y∈Y

and so φ is bounded, that is, φ ∈ b[X ×Y, Z]. Thus if X is a Banach space and φy ∈ B[X , Z], then (c) holds. A symmetrical argument shows that if Y is  a Banach space and φx ∈ B[Y, Z], then (c) holds as well. As is readily verified, if φ ∈ b[X ×Y, Z], then φ =

sup x=y=1

φ(x, y) =

sup x≤1,y≤1

φ(x, y) =

sup x P  = 1, by taking P = E or P = F . (b) Another example with a similar argument. Let X be a Hilbert space and consider the Banach algebras B[X ] and Z = B[X ] ⊕ B[X ], where Z consists of A O all orthogonal direct sums A ⊕ B = O B of A, B in B[X ]. Take the projection E ∈ B[Z] given by E(A ⊕ B) = O ⊕ (A + B) for A, B ∈ B[X ], so that E > 1. Set M = R(E) = {0} ⊕ B[X ] ⊆ B[X ] ⊕ B[X ] = Z. Consider the restriction of E to M, E|M = IM : M → Z, which acts as the identity IM on M. Now take φ ∈ b[M×M, Z ] defined by φ = IM ·IM and let φ ∈ b[Z×Z, Z] be defined by φ = I ·I where I ∈ B[Z] denotes the identity on Z. In other words, φ is given by φ(O ⊕ B1 , O ⊕ B2 ) = (O ⊕ B1 )·(O ⊕ B2 ) = O ⊕ (B1 ·B2 ) for every  1 ⊕ B1 , A2 ⊕ B2 ) = (A1 ⊕ B1 )·(A2 ⊕B2 ) = B1 , B2 ∈ B[X ], and φ is given by φ(A  M×M = IM ·IM = φ (A1 ·A2 ) ⊕ (B1 ·B2 ) for A1 , A2 , B1 , B2 in B[X ]. Thus φ|   / E = P = 1 and φ = IM ·IM  = 1 = I ·I = φ. Hence φ = φ =⇒ (with P ∈ B[Z] taken in a similar fashion as P was taken in item (a) above). As expected, if X and Y are Hilbert spaces, then we get a full extension result for bounded bilinear maps with no restrictions on the linear manifolds M and N , which is stated in Corollary 6.10 below. In particular, with Z = F we get a Hahn–Banach-like theorem for bounded bilinear functionals on the Cartesian product of linear manifolds of Hilbert spaces.

6.3 Extension of Bounded Bilinear Maps

127

Corollary 6.10. Every bounded bilinear map φ : M×N → Z defined on the Cartesian product of arbitrary linear manifolds M and N of arbitrary Hilbert spaces X and Y into an arbitrary Banach space Z has a bounded bilinear  = φ. extension φ : X ×Y → Z over X ×Y such that φ Proof. (a) If X is a Hilbert space, then M− + M⊥ = X for every linear manifold M of X by the Projection Theorem (cf. Proposition 5.Q), where M− ∩ M⊥ = {0}. So every subspace of a Hilbert space is complemented (cf. Remark 5.3). Thus Theorem 6.8(a) applies to every linear manifold M of a Hilbert space X and to every linear manifold N of a Hilbert space Y. (b) If X and Y are Hilbert spaces, then take the orthogonal projections E in B[X ] with R(E) = M− and P in B[Y] with R(P ) = N −, which are such that  = φ by Theorem 6.8(b).  E = P  = 1 (cf. Proposition 5.R), and so φ Extensions of linear transformations are not unique (cf. example in Remark 1.2). Thus extensions of bilinear maps are not unique by Theorem 2.4. If M and N are ranges or kernels of operators in SΓ [X ] ⊆ B[X ] and in SΓ [Y] ⊆ B[Y] (as defined in the paragraph before Corollary 5.7) respectively, then we get the following extension result for bounded bilinear maps. Corollary 6.11. Let X and Y be Banach spaces. Let M and N be linear manifolds of X and Y for which there are operators T ∈ SΓ [X ] ⊆ B[X ] and S ∈ SΓ [Y] ⊆ B[Y] such that M− = R(T )− or M− = N (T )

and

N − = R(S)− or N − = N (S),

according to whether T lies in ΓR [X ] or ΓN [X ] and S lies in ΓR [Y] or ΓN [Y]. If φ : M×N → Z is a bounded bilinear map into an arbitrary Banach space Z, then there exists a bounded bilinear extension φ : X ×Y → Z of φ over X ×Y. Proof. As in the proof of Corollary 5.7, this follows immediately from Theorem 6.8. If T ∈ SΓ [X ] and S ∈ SΓ [Y], then either R(T )− or N (T ) is complemented in X and either R(S)− or N (S) is complemented in Y. If M− = R(T )− or M− = N (T ) and if N − = R(S)− or N − = N (S), then φ ∈ b[M×N , Z] for any Banach space Z has an extension φ ∈ b[X ×Y, Z] by Theorem 6.8.  Take arbitrary nonzero operators T ∈ B[X ] and S ∈ B[Y], and take an arbitrary bounded bilinear map φ ∈ b[R(T )×R(S), Z]. Now define a map ψ : X ×Y → Z by ψ(x, y) = φ(T x, Sy) for every (x, y) ∈ X ×Y. According to Proposition 6.B below, ψ ∈ b[X ×Y, Z]. If T and S are projections, then ψ is an extension of φ. (Indeed, since ψ|R(T )×R(S) = φ(T |R(T ) , S|R(S) ), if T and S are projections, or equivalently, if T |R(T ) = IR(T ) , identity on R(T ), and S|R(S) = IR(S) , identity on R(S), then ψ|R(T )×R(S) = φ.) However, if there  R(T )×R(S) = φ (e.g., under the assumptions of is a φ ∈ b[X ×Y, Z] such that φ|  Corollary 6.10), then φ(T x, Sy) = φ(T x, Sy) = ψ(x, y) for every (x, y) ∈ X ×Y.

128

6. Bounded Bilinear Maps

6.4 Additional Propositions Let X , Y, Z be nonzero normed spaces over the same field F , let b[X ×Y, Z] and L[X , Z] denote the linear space of all bilinear maps from X ×Y to Z and of all linear transformations of X into Z, and let b[X ×Y, Z] and B[X , Z] denote the normed spaces of all bounded (i.e., continuous) bilinear maps from X ×Y to Z and of all bounded linear transformations of X into Z. Proposition 6.A. If X and Y are finite-dimensional normed spaces, then b[X ×Y, Z] = b[X ×Y, Z]. Proposition 6.B. Let X , X , Y, Y , Z, Z  be normed spaces (over the same field ). Take bounded linear transformations A ∈ B[X , X ], B ∈ B[Y , Y] and map φ ∈ C ∈ B[Z, Z  ], and take a bounded bilinear   b[X ×Y, Z]. The mapping ψ : X  ×Y  → Z  given by ψ(x, y  ) = C φ(Ax, By  ) for every (x , y  ) ∈ X  ×Y  is a bounded bilinear map. Actually, ψ = C ◦ φ(A, B) ∈ b[X  ×Y , Z  ]

with

ψ ≤ φABC.

Proposition 6.C. The evaluation map φ : B[X , Y]×X → Y given by φ(T, x) = T x

for every

(T, x) ∈ B[X , Y]×X

is a surjective bounded bilinear map with norm one. Recall: all nonzero bilinear forms (i.e., for Z = F ) are trivially surjective. Proposition 6.D. Let Z be a normed algebra. Take any bounded linear transformations S ∈ B[X , Z] and T ∈ B[Y, Z]. The map φ : X ×Y → Z defined by φ(x, y) = Sx · T y

for every

(x, y) ∈ X ×Y

is a bounded bilinear map with φ ≤ ST . In particular, for fixed bounded linear functionals f ∈ X ∗ and g ∈ Y ∗ the bilinear form φ : X ×Y → F given by φ(x, y) = f (x) · g(y) for every (x, y) ∈ X ×Y is bounded with φ = f g. There is no Hahn–Banach Theorem and no uniform continuity for bounded bilinear maps (cf. Remark 6.4(b,c)) but there is extension over completion (cf. Theorem 6.5). The next proposition shows that there is no Open Mapping Theorem for surjective bounded bilinear maps of the Cartesian product of Banach spaces onto a Banach space. Proposition 6.E. Let Z be a finite-dimensional normed space over a field F of dimension greater than one. The scalar multiplication F ×Z → Z (given by (α, z) → α z) is a surjective bilinear map but it is not an open mapping. The Closed Graph Theorem, however, travels well from linear transformations to bilinear maps. The graph Gφ of a bilinear map φ : X ×Y → Z,

6.4 Additional Propositions

129

  Gφ = ((x, y), z) ∈ (X ×Y)×Z : z = φ(x, y)   = (x, y, φ(x, y)) ∈ X ×Y×Z : for every (x, y) ∈ X ×Y , is a subset of the Cartesian product (X ×Y)×Z = X ×Y×Z. If X , Y, Z are normed spaces, then equip the Cartesian product with a product metric d generated by their norms (cf. Section 5.1 — as in Proposition 5.D in Section 5.4). The Closed Graph Theorem for bilinear maps is given below. Proposition 6.F. If X , Y, Z are Banach spaces and φ : X ×Y → Z is bilinear, then φ is continuous if and only if Gφ is closed . As for the Banach–Steinhaus Theorem, its consequence ensuring a bounded limit for pointwise convergence of sequences of bounded mappings, which holds for sequences of bounded linear transformations (as in Proposition 5.G), also holds for sequences of bounded bilinear maps (as in Proposition 6.H below). Proposition 6.G. Let {φ(ω)}ω∈Ω be an indexed family of bounded bilinear maps φ(ω) of the Cartesian product X ×Y of Banach spaces X and Y into a normed space Z. If supω∈Ω φ(ω)(x, y) < ∞ for every (x, y) ∈ X ×Y, then supω∈Ω φx (ω) < ∞ for each x ∈ X and supω∈Ω φy (ω) < ∞ for each y ∈ Y. A sequence of bilinear maps of the Cartesian product X ×Y of normed spaces X and Y into a normed space Z (i.e., a b[X ×Y, Z]-valued sequence) {φ(n)} is pointwise convergent if for every pair (x, y) in X ×Y the Z-valued sequence {φ(n)(x, y)} converges in Z to, say, z(x,y) in Z. Since the limit is unique, this defines a map φ : X ×Y → Z given by φ(x, y) = z(x,y) for every (x, y) in X ×Y. It is readily verified that φ is bilinear. Thus a b[X ×Y, Z]valued sequence {φ(n)} is pointwise convergent if and only if there exists a map φ in b[X ×Y, Z] such that (φ(n) − φ)(x, y) → 0 for every (x, y) in X ×Y. Now consider a sequence {φ(n)} of bounded bilinear maps of X ×Y into Z (i.e., a b[X ×Y, Z]-valued sequence). An application of the Banach–Steinhaus Theorem for bounded linear transformations ensures that if X and Y are Banach spaces and {φ(n)} is pointwise convergent, then the limit φ is bounded. The next result is the bilinear counterpart of Proposition 5.G. Since Proposition 5.G is a consequence of Proposition 5.F, the proposition below is regarded as the Banach–Steinhaus Theorem for bilinear maps. Proposition 6.H. If X and Y are Banach spaces and Z is a normed space, then a b[X ×Y, Z]-valued sequence {φ(n)} is pointwise convergent if and only if there is a map φ ∈ b[X ×Y, Z] such that (φ(n) − φ)(x, y) → 0 for every pair (x, y) ∈ X ×Y (i.e., pointwise convergence implies a bounded limit). Recall that a function between topological spaces is continuous if and only if it preserves convergent nets (i.e., if and only if f (xα ) → f (x) whenever {xα } is a net converging to x). For metric spaces, nets can be replaced by sequences. A function between topological spaces is sequentially continuous if it preserves convergent sequences. Thus continuity implies sequential continuity, and the

130

6. Bounded Bilinear Maps

concepts coincide in a metric space. The weak topology of (infinite-dimensional) normed spaces is not metrizable. Along these lines, also recall that a linear functional on a normed space X is weakly continuous (i.e., continuous with respect to the weak topology of X ) if and only if it is strongly continuous (i.e., continuous with respect to the norm topology of X ). This also does not survive when transferred from linear functionals to bilinear functionals. Proposition 6.I. Continuous bilinear functionals on the Cartesian product of Banach spaces may not be weakly continuous. Let Z ∗∗ = (Z ∗ )∗ denote the second dual (or bidual ) of a normed space Z. The next proposition states a significant fact that will be discussed below. Proposition 6.J. If X and Y are Banach spaces, then every bounded bilinear  = φ. form φ ∈ b[X ×Y, F ] has an extension φ ∈ b[X ∗∗ ×Y ∗∗ , F ] with φ Sesquilinear forms σ : X×Y → F were defined in Section 2.4 for linear spaces X and Y. Inner products on a linear space X were defined in Section 5.4 as sesquilinear forms · ; · : X ×X → F generating a quadratic form  ·  : X → R . Inner product spaces and Hilbert spaces were defined in Section 5.4. The notion of completion of inner product spaces is similar to that of normed spaces. If the image of a linear isometry on an inner product space X is dense in a Hilbert space X, then X is a completion of X . Equivalently, if an inner product space X is unitarily equivalent to a dense linear manifold of a Hilbert space X, then X is a completion of X . Inner product spaces, being normed spaces, satisfy in particular the assumptions of Theorem 5.1. The remaining question is whether the Banach spaces obtained by completion of inner product spaces are Hilbert spaces. Yes, they are. So completion of inner product spaces agrees with the above definition. The counterpart of Theorem 5.1 for inner product spaces reads as follows. Every inner product space has a completion. Two completions of an inner product space are unitarily equivalent. If X and Y are completions of inner product spaces X and Y, then every T ∈ B[X , Y] has an exten over the completion X of X into the completion Y of sion T ∈ B[X, Y]  Y. Also, T is unique up to unitary transformations, and T = T . (See, e.g., [52, Definition 5.22 and Theorem 5.23].) Moreover, the inner product equipping the Hilbert space X (completion of X ) is precisely the extension over completion of the inner product space equipping X . This is the sesquilinear version of the extension over completion for bilinear forms in Theorem 6.5. Proposition 6.K. Let X be the completion of an inner product space X . The inner product · ; ·X : X ×X → F has an extension over completion on X×X, · ; ·X : X×X → F which is unique up to unitary equivalence. If x , y ∈ X are extensions of x, y ∈ X , then  x ; yX = x ; yX .

6.4 Additional Propositions

131

Notes: Bilinear maps in a linear-space setting, considered in Chapter 2, played a central role in defining tensor products of linear spaces as elaborated in Chapter 3. The next chapter will consider tensor products of normed spaces, where the tensor product space will be equipped with a suitable norm, and this naturally requires the notion of bounded bilinear maps in a normed-space setting. Proposition 6.A is an immediate consequence of Theorem 6.2 since the identities B[X , Z] = L[X , Z] and B[Y, Z] = L[Y, Z] apply to the linear transformations φy and φx whenever X and Y are finite-dimensional, since every finite-dimensional normed space is a Banach space, and since linear transformations on finite-dimensional normed spaces are continuous (see, e.g., [52, Corollaries 4.28 and 4.30]). The bilinearity in Proposition 6.B follows from Remark 2.2 and boundedness (as well as the norm inequalities) from Theorem 6.1. The evaluation map φ of Proposition 6.C is bilinear (cf. Proposition 2.H), surjective (indeed, if y ∈ Y and f ∈ X ∗, then T x = f (x)y defines T ∈ B[X , Y] for which y ∈ R(T )), clearly contractive (i.e., φ ≤ 1), and by setting T (·) = f (·)y with f  = y = 1 it follows that φ = 1. Proposition 6.D is straightforward in light of the pertinent definitions, where Sx·T y ≤ SxT y since Z is a normed algebra, and the norm identity for the scalar case holds because |f (x)·g(y)| = |f (x)||g(y)|. For Proposition 6.E see, e.g., [15, Section 1.7]. Proof of Proposition 6.F. Consider the normed space obtained by the direct sum (X ⊕ Y ⊕ Z,  · ) = (X ×Y×Z, ⊕,  , F ,  · ) where the norm  ·  on the linear space X ⊕ Y ⊕ Z is the one generating the product metric d on the Cartesian product X ×Y×Z. Suppose the graph Gφ of φ is a closed subset of (X ⊕ Y ⊕ Z,  · ). Take the (closed) subspace M = {(0, y, z) ∈ X ⊕ Y ⊕ Z : (y, z) ∈ Y ⊕ Z} of (X ⊕ Y ⊕ Z,  · ). Identify each x in X with (x, 0, 0) in X ⊕ Y ⊕ Z. Consider the translation [x] = M ⊕ x for each x ∈ X , again a closed subset of (X ⊕ Y ⊕ Z,  · ). Thus the intersection Gφ ∩ [x] is closed. The graph of each linear transformation φx ∈ L[Y, Z],   Gφx = (y, z) ∈ Y ⊕ Z : z = φx (y) = (Gφ ∩ [x]) ⊕ (−x) ⊆ Y ⊕ Z, regarded as a subset of X ⊕ Y ⊕ Z, is the translation of a closed set, thus closed in (X ⊕ Y ⊕ Z,  · ), and naturally (and isometrically) embedded as a closed set in the subspace (Y ⊕ Z,  · ) ∼ = (M,  · ). Then by the Closed Graph Theorem for linear transformations (Proposition 5.D), each φx is continuous. Symmetrically, each φy is continuous. Therefore φ is continuous by Theorem 6.2. The converse is readily verified by the Closed Set Theorem.  (Indeed, suppose φ is continuous. Take any Gφ -valued convergent sequence {(xn , yn , φ(xn , yn ))}, converging in (X ×Y×Z, d) to (x, y, z) ∈ X ×Y×Z. Since d is a product metric, the X -valued, the Y-valued, and the Z-valued sequences {xk }, {yk }, {φ(xn , yn )} converge in (X ,  · ), (Y,  · ), (Z,  · ) to x ∈ X , y ∈ Y, and z ∈ Z, respectively. Since φ is continuous, it preserves convergence. By uniqueness of the limit, φ(xn , yn ) → φ(limn xn , limn yn ) = φ(x, y) = z in (Z,  · ). Thus (x, y, z) lies in Gφ and hence Gφ is closed in the metric space (X ×Y×Z, d) by the Closed Set Theorem (see, e.g., [52, Theorem 3.30]).)

132

6. Bounded Bilinear Maps

Proof of Proposition 6.G. Since φ(ω)(x, y) = φx (ω)(y) = φy (ω)(x) for every (ω, x, y) ∈ Ω×X ×Y, it follows that if supω∈Ω φ(ω)(x, y) < ∞ for every (x, y) ∈ X ×Y, then for each x ∈ X we get supω∈Ω φx (ω)(y) < ∞ for every y ∈ Y, and for each y ∈ Y we get supω∈Ω φy (ω)(x) < ∞ for every x ∈ X . Now apply the Banach–Steinhaus Theorem (Proposition 5.F) to the bounded  linear transformations φx (ω) ∈ B[Y, Z] and φy (ω) ∈ B[X , Z]. Proof of Proposition 6.H. Suppose a b[X ×Y, Z]-valued sequence {φ(n)} converges pointwise to φ ∈ b[X ×Y, Z]. This means (φ(n) − φ)(x, y) → 0 for every (x, y) ∈ X ×Y, which implies φ(n)(x, y) → φ(x, y), which in turn implies supn φ(n)(x, y) < ∞ for every (x, y) ∈ X ×Y. Thus supn φx (n) < ∞ for every x ∈ X by Proposition 6.G. Moreover, for every (x, y) ∈ X ×Y, φx (y) = φ(x, y) = limn φ(n)(x, y) = limn φx (n)(y) ≤ supn φx (n)y. x (y) So φx  = supy=0 φy ≤ supn φx (n) < ∞ for each x ∈ X . Symmetrically, y φ  < ∞ for each y ∈ Y. Thus φx ∈ B[Y, Z] and φy ∈ B[X , Z] for every x and y in the Banach spaces X and Y. Then φ ∈ B[X ×Y, Z] by Theorem 6.2. 

Proof of Proposition 6.I. The inner product on the real Hilbert space 2+ (of all square-summable sequences of real numbers) is a bilinear form which is trivially continuous (by the Schwartz inequality) but it is not weakly sequentially continuous. In fact, two 2+ -valued sequences may individually converge weakly on 2+ but the scalar sequence of their inner products may not converge. For instance, the sequence of unit sequences {en } converges weakly to zero (since en ; x → 0 for every x ∈ 2+ ) and en ; en  = 1 for all positive integers n.  Proposition 6.J is a particularly important result on bilinear extension to biduals. It is based on the following classical result (which is another consequence of the Hahn–Banach Theorem). Every normed space Z is isometrically embedded in its second dual Z ∗∗. Indeed, the natural embedding Φ : Z → Z ∗∗ sets an isometric isomorphism of Z onto its range Φ(Z) ⊆ Z ∗∗. The isometric embedding Φ is defined by Φ(z) = ζz ∈ Z ∗∗ for every z ∈ Z, with ζz given by ζz (h) = h(z) for every h ∈ Z ∗, for each z ∈ Z (see, e.g., [52, Theorem 4.66]). We drop the subscript and denote ζz ∈ Φ(Z) by ζ for simplicity. Hence h ; ζ = ζ(h) = h(z) = z ; h

for every

h ∈ Z∗

(∗)

establishes the identification z ↔ ζ between z ∈ Z and ζ ∈ Φ(Z) ⊆ Z ∗∗. So Z can be regarded as a linear manifold of Z ∗∗ where Z is identified with the subspace Φ(Z) of Z ∗∗ (i.e., Z is identified with the range Φ(Z) of the natural embedding Φ of Z into Z ∗∗ ). Proof of Proposition 6.J. Let X and Y be arbitrary nonzero Banach spaces. Identify X and Y as subspaces of X ∗∗ and Y ∗∗. Take an arbitrary bounded bilinear form φ ∈ b[X ×Y, F ]. According to Theorem 6.3, φ ∼ = T for T ∈ B[X , Y ∗ ] given by (T x)(·) = φx (·) : Y → F so that

6.4 Additional Propositions

133

φ(x, y) = (T x)(y) = y ; T x for every (x, y) ∈ X ×Y. Consider the (normed-space) adjoint T ∗ ∈ B[Y ∗∗, X ∗ ] of T ∈ B[X , Y ∗ ]. Take a form φ : X ∗∗ ×Y ∗∗ → F defined by  η) = ϕ(T ∗ η) = T ∗ η ; ϕ φ(ϕ, for every (ϕ, η) ∈ X ∗∗×Y ∗∗. It is clear that the form φ is bilinear and bounded.  Φ(X )×Φ(Y) of φ to the Cartesian product Φ(X )×Φ(Y) Consider the restriction φ| of subspaces Φ(X ) of X ∗∗ and Φ(Y) of Y ∗∗, which are identified with X and Y, respectively. Therefore when acting on Φ(X )×Φ(Y), and identifying each pair (ϕ, η) ∈ Φ(X )×Φ(Y) with the unique pair (x, y) ∈ X ×Y, we write (by using (∗) twice, and according to the definition of adjoint)  y) = φ(ϕ,  η) = T ∗ η ; ϕ = x ; T ∗ η = T x ; η = y ; T x = φ(x, y) φ(x, for every (x, y) ∈ X ×Y. Thus, identifying Φ(X ) with X and Φ(Y) with Y,  X ×Y . φ = φ| Moreover, since φ ∼ = T,  = φ

sup

 Φ(X )×Φ(Y)  ≤ φ.  φ(T ∗ η) ≤ T ∗  = T  = φ = φ| 

ϕ=1,η=1

(It is worth noticing en passant that for an arbitrary φ ∈ b[X ×Y, Z] there is an adjoint φ∗ ∈ b[Z ∗ ×X, Y ∗ ] of φ, given by φ∗ (f, x) = f (φ(x, y)) for every f ∈ Z ∗ , x ∈ X and y ∈ Y, whose third iteration φ∗∗∗ ∈ b[X ∗∗ ×Y ∗∗, Z ∗∗ ] yields a norm-preserving extension of φ [1, p. 839] — as φ ∈ b[X ∗∗ ×Y ∗∗, F ] in Proposition 6.J yields a norm-preserving extension of φ ∈ b[X ×Y, F ].) Proof of Proposition 6.K. Let  · X be the norm on X induced by the inner product · ; ·X on X . It can be shown (see, e.g., [52, proof of Theorem 5.23]) that the norm  · X on the completion X of X satisfies the parallelogram law (see, e.g., [52, Proposition 5.4]). This is equivalent to saying that the norm  · X is generated by an inner product · ; ·X (see, e.g., [52, Theorem 5.5]). Also, uniqueness comes naturally from the definition of completion. Moreover, according to the completion procedure (see Section 5.1), let x  ∈ X be the correspondent (i.e., the extension) of x ∈ X so that  xX = xX . Using the polarization identity (see, e.g., [52, Proposition 5.4]), and recalling that x +y =x  + y, we get the inner product identity from the norm identity:  x ; yX = x ; yX .



Suggested Readings Brown and Pearcy [5] Defant and Floret [15]

Greub [27] Ryan [83]

7 Norms on Tensor Products

This chapter contains a detailed account of basic results on tensor products of Banach spaces. It focuses on suitable norms equipping a tensor product space. Essentially, the attributes belonging to a norm on a tensor product space (of a pair of normed spaces over the same field) that make it suitable (the proper term will be “reasonable”) are the properties (a) and (b) in Theorem 7.2.

7.1 Reasonable Crossnorms Usually we adopt a generic notation  ·  for norms on normed spaces, indexed if necessary as we agreed in Section 5.1. Norms on tensor product spaces, however, will always be indexed. If the linear space X ⊗ Y is equipped with a norm  · α , then we write X ⊗α Y for the normed space (X ⊗ Y,  · α ). Let  · ∗α be the norm on the dual (X ⊗α Y)∗ when X ⊗ Y is equipped with the norm  · α (i.e.,  · ∗α is the induced uniform norm on B[X ⊗α Y, F ]). Summing up: X ⊗α Y = (X ⊗ Y,  · α ) When necessary we write  · X

and ⊗α Y

(X ⊗α Y)∗ = (B[X ⊗α Y, F ],  · ∗α ). for  · α and  · (X

⊗α Y)∗

for  · ∗α .

Definition 7.1. Let X and Y be normed spaces. A norm  · α on the linear space X ⊗ Y is a reasonable crossnorm (sometimes denoted by α(·)) if, for every x ∈ X , y ∈ Y, f ∈ X ∗, g ∈ Y ∗, (a) x ⊗ yα ≤ x y, (b) f ⊗ g lies in (X ⊗α Y)∗ , and f ⊗ g∗α ≤ f  g. Definition 7.1 imposes conditions on the norms of single tensors x ⊗ y and f ⊗ g. The first basic observations about it are: (i) the above inequalities are in fact identities (as we will see next), and (ii) a reasonable crossnorm is such that the inclusion involving algebraic duals in Remark 3.18(b), viz., X  ⊗ Y  ⊆ (X ⊗ Y) , is sharpened to X ∗ ⊗ Y ∗ ⊆ (X ⊗α Y)∗ . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Kubrusly, Bilinear Maps and Tensor Products in Operator Theory, Universitext, https://doi.org/10.1007/978-3-031-34093-2 7

135

136

7. Norms on Tensor Products

Theorem 7.2. Let X and Y be normed spaces. A norm  · α on X ⊗ Y is a reasonable crossnorm if and only if, for every x ∈ X , y ∈ Y, f ∈ X ∗, g ∈ Y ∗, (a) x ⊗ yα = x y, (b) f ⊗ g lies in (X ⊗α Y)∗ , and f ⊗ g∗α = f  g. Proof. Consider the setup of Definition 7.1 where  · ∗α denotes the (induced uniform) norm on (X ⊗ Y)∗ when X ⊗ Y is equipped with the norm  · α . If the identities in (a) and (b) hold, then the respective inequalities in Definition 7.1 hold tautologically. The converse goes as follows. Suppose  · α is a reasonable crossnorm on X ⊗ Y as in Definition 7.1. (a) Take arbitrary x ∈ X and y ∈ Y. The Hahn–Banach Theorem ensures the existence of f ∈ X ∗ and g ∈ Y ∗ with f  = g = 1 for which |f (x)| = x and |g(y)| = y (see, e.g., [52, Corollary 4.64]). Thus f ⊗ g∗α ≤ 1 by Definition 7.1(b), and so (since (f ⊗ g)(x ⊗ y) = f (x) g(y) — cf. Remark 3.18(b)) x y = |f (x) g(y)| = |(f ⊗ g)(x ⊗ y)| ≤ f ⊗ g∗α x ⊗ yα ≤ x ⊗ yα . Hence x ⊗ yα = x y by Definition 7.1(a). (b) Take any f ∈ X ∗ and g ∈ Y ∗. As x ⊗ yα ≤ x y by Definition 7.1(a), (x)| f  g = sup |fx sup |g(y)| y = sup x=0

≤ sup

x⊗y=0

y=0

|(f ⊗g) (x⊗y)| x⊗yα

x,y=0

≤

|f (x) g(y)| xy

sup

0=∈X ⊗Y

= sup

|(f ⊗g)| α

x,y=0

|(f ⊗g) (x⊗y)| xy

= f ⊗ g∗α .

Hence f ⊗ g∗α = f  g by Definition 7.1(b).

 

Thus a tensor product space X ⊗ Y equipped with a reasonable crossnorm  · α yields the normed space X ⊗α Y = (X ⊗ Y,  · α ), and its dual (X ⊗α Y)∗ = (X ⊗ Y,  · α )∗ = B[(X ⊗ Y,  · α ), F ] = B[X ⊗α Y, F ] is equipped with  · ∗α , which stands for the induced uniform norm on the Banach space B[X ⊗α Y, F ]. By Definition 7.1, every single tensor f ⊗ g in the space X ∗ ⊗ Y ∗ lies in (X ⊗α Y)∗. Then every element Ω =  tensor product ∗ ∗ ∗ ∗ ∗ i fi ⊗ gi in X ⊗ Y lies in (X ⊗α Y) . In fact, the linear space X ⊗ Y is ∗ an algebraic linear manifold of the linear space (X ⊗α Y) , X ∗ ⊗ Y ∗ ⊆ (X ⊗α Y)∗ , whenever the tensor product space X ⊗ Y is equipped with any reasonable crossnorm  · α . The above linear-space inclusion follows the pattern X  ⊗ Y  ⊆ (X ⊗ Y) of Remark 3.18(b), and the next linear-space inclusions are clear: X ∗ ⊗ Y∗ ⊆ X  ⊗ Y

and

(X ⊗α Y)∗ ⊆ (X ⊗ Y) .

Thus what is required in Definition 7.1(b) is that every single tensor

7.1 Reasonable Crossnorms

f ⊗g

137

X ∗ ⊗ Y ∗ ⊆ X  ⊗ Y  ⊆ (X ⊗ Y)

in

lies in (X ⊗α Y)∗. So besides lying in (X ⊗ Y) it is continuous as a functional from the normed space X ⊗α Y = (X ⊗ Y,  · α ) (equipped with any reasonable crossnorm  · α ) to F (equipped with its usual norm | · |). Since  · ∗α is the norm on (X ⊗α Y)∗ = B[X ⊗α Y, F ], and X ∗ ⊗ Y ∗ ⊆ (X ⊗α Y)∗ , we get f ⊗ g∗α = f ⊗ g(X

⊗α Y)∗

= f ⊗ gB[X

⊗α Y, F ]

= f  g

according to Theorem 7.2. In general, for every Λ ∈ (X ⊗α Y)∗ , Λ∗α = Λ(X =

⊗α Y)∗

sup

0=∈X ⊗Y

|Λ()| α

In particular, for every Ω = Ω∗α =

sup

0=∈X ⊗Y

|Ω()| α

=

= ΛB[X

 j

=

⊗α Y,F ]

sup

=

0=Σi xi ⊗yi ∈X ⊗Y

sup

∈X ⊗Y, α ≤1

|Λ()|

   Λ i xi ⊗yi    .  i xi ⊗yi  α

fj ⊗ gj ∈ X ∗ ⊗ Y ∗ ⊆ (X ⊗α Y)∗,

sup

   j fj ⊗gj ()

0=∈X ⊗Y

α

=

0=Σi

    i,j fj (xi ) gj (yi )    sup .  i xi ⊗yi  x ⊗y ∈X ⊗Y i

i

α

∗

∗

∗

More particularly, for every single tensor f ⊗ g ∈ X ⊗ Y ⊆ (X ⊗α Y) ,        f ⊗g i xi ⊗yi  f (xi ) g(yi ) i     f ⊗ g∗α = sup = sup i xi ⊗yi  i xi ⊗yi  . 0=Σ x ⊗y ∈X ⊗Y 0=Σ x ⊗y ∈X ⊗Y i

i

i

α

i

i

i

α

However, by Theorem 7.2 again, f ⊗ g∗α = f ⊗ gβ = f  g for every single tensor f ⊗ g ∈ X ∗ ⊗ Y ∗, for every reasonable crossnorm  · α on X ⊗ Y, and for every reasonable crossnorm  · β on X ∗ ⊗ Y ∗ . Theorem 7.3. Let  · α be a reasonable crossnorm on a tensor product space X ⊗ Y of normed spaces X and Y. Take the dual (X ⊗α Y)∗ of X ⊗α Y = (X ⊗ Y,  · α ). When restricted to X ∗ ⊗ Y ∗, the norm  · ∗α on (X ⊗α Y)∗ is a reasonable crossnorm on X ∗ ⊗ Y ∗ . In other words, since X ∗ ⊗ Y ∗ ⊆ (X ⊗α Y)∗, when restricted to the tensor product space X ∗ ⊗ Y ∗ the norm  · ∗α on the Banach space (X ⊗α Y)∗ is such that for arbitrary single tensors f ⊗ g in X ∗ ⊗ Y ∗ and ϕ ⊗ η in X ∗∗ ⊗ Y ∗∗, (a) f ⊗ g∗α ≤ f  g, (b) ϕ ⊗ η lies in (X ∗ ⊗∗α Y ∗ )∗ so that X ∗∗ ⊗ Y ∗∗ ⊆ (X ∗ ⊗∗α Y ∗ )∗ , and ϕ ⊗ η∗∗α ≤ ϕη (where  · ∗∗α is the norm on (X ∗ ⊗∗αY ∗ )∗ when X ∗ ⊗ Y ∗ is equipped with the norm  · ∗α ), and the inequalities in (a) and (b) are in fact identities (by Theorem 7.2).

138

7. Norms on Tensor Products

Proof. Let  · α be a reasonable crossnorm on X ⊗ Y. Take the normed space X ⊗α Y. Let  · ∗α be the norm on the dual of X ⊗α Y ; that is, the induced uniform norm on (X ⊗α Y)∗ = B[X ⊗α Y, F ]. Since X ∗ ⊗ Y ∗ ⊆ (X ⊗α Y)∗ , equip X ∗ ⊗ Y ∗ with the norm  · ∗α inherited from (X ⊗α Y)∗ . So X ∗ ⊗∗α Y ∗ is a linear manifold of (X ⊗α Y)∗ . Let  · ∗∗α be the norm on the dual of X ∗ ⊗∗α Y ∗ ; that is, the induced uniform norm on (X ∗ ⊗∗α Y ∗ )∗ = B[X ∗ ⊗∗α Y ∗ , F ]. The inequality in (a) holds according to Definition 7.1(b). Take an arbitrary single tensor ϕ ⊗ η in X ∗∗ ⊗ Y ∗∗, so that ϕ ∈ X ∗∗ = B[X ∗, F ] and η ∈ Y ∗∗ = B[Y ∗, F ]. Since (cf. Remark 3.18(b) again) X ∗∗ ⊗ Y ∗∗ ⊆ X ∗  ⊗ Y ∗  ⊆ (X ∗ ⊗ Y ∗ ) , then ϕ ⊗ η lies in (X ∗ ⊗ Y ∗ ) . Thus ϕ ⊗ η : X ∗ ⊗ Y ∗ → F is a linear functional. We apply the next claim to verify that the assertions in (b) hold as well. Claim. Let X and Y be normed spaces. Take arbitrary nonzero ϕ ∈ X ∗∗ and η ∈ Y ∗∗ . There are nets {xγ }γ∈Γ and {yγ  }γ  ∈Γ  of elements in X and Y with xγ  ≤ ϕ

and

yγ   ≤ η,

such that, for every f ∈ X ∗ and g ∈ Y ∗, lim γ f (xγ ) = ϕ(f )

and

lim γ  g(yγ  ) = η(g).

Proof of Claim. For each x ∈ X consider the functional ϕx : X ∗ → F given by ϕx (f ) = f (x) for every f ∈ X ∗, which is linear and bounded (i.e., ϕx ∈ X ∗∗ ) with ϕx  = x. Let Φ : X → X ∗∗ be defined by Φ(x) = ϕx , so that Φ(x)(f ) = ϕx (f ) = f (x) for every x ∈ X and every f ∈ X ∗. This Φ is an isometric isomorphism of X onto R(Φ) = Φ(X ) ⊆ X ∗∗, which is the natural embedding of X into X ∗∗ (cf. proof of Proposition 6.J in Section 6.4). The Goldstine Theorem (see, e.g., [70, Theorem 2.6.26]) says that Φ(BX ) ⊂ X ∗∗ is weakly* dense in BX ∗∗ ⊂ X ∗∗ (where BX and BX ∗∗ are the closed unit balls in X and in X ∗∗ ). (Recall that a set S is dense in a topological space X if and only if for every point in X there is an S-valued net converging to it (see, e.g., [12, Proposition A.2.2]). Thus every ϕ in BX ∗∗ ⊂ X ∗∗ is the weak* limit of a Φ(BX )-valued net {Φ(xγ )}γ∈Γ in the dual X ∗∗ of X ∗, where {xγ }γ∈Γ is a BX -valued net of w∗ ϕ, which means Φ(xγ )(f ) → ϕ(f ) for every elements from X (i.e., Φ(xγ ) −→ ∗ f ∈ X ). In other words, every functional ϕ in BX ∗∗ ⊂ X ∗∗ is such that ϕ(f ) = limγ Φ(xγ )(f ) for every functional f ∈ X ∗.) ϕ ϕ Therefore, for every 0 = ϕ ∈ X ∗∗, set ϕ ∈ BX ∗∗ ⊂ X ∗∗ (with  ϕ  = 1) so that there exists an X -valued net {xγ }γ∈Γ with xγ  ≤ ϕ (which implies xγ  xγ  ∗ ϕ ≤ 1 and hence ϕ ∈ BX ⊂ X ) such that for every f ∈ X

7.1 Reasonable Crossnorms ϕ ϕ (f )

x

γ = limγ Φ( ϕ )(f ),

which means

ϕ ϕ (f )(x)

139

x

γ = limγ f ( ϕ ),

and so lim γ f (xγ ) = ϕ(f ). Similarly, yγ  ≤ η and lim γ  g(yγ  ) = η(g).    ∗ ∗ Take an arbitrary Ω = i fi ⊗ gi from X ⊗ Y . Thus by the above claim, and since the norm  · α on X ⊗ Y is a reasonable crossnorm,          |(ϕ ⊗ η)Ω| = (ϕ ⊗ η) fi ⊗ gi  =  ϕ(fi ) η(gi ) i i         = lim γ fi (xγ ) lim γ  gi (yγ  ) = lim γ,γ   fi (xγ ) gi (yγ  ) i i         = lim γ,γ   (fi ⊗ gi )(xγ ⊗ yγ  ) = lim γ,γ  Ω(xγ ⊗ yγ  ) i

≤ lim sup γ,γ  Ω∗α xγ ⊗ yγ  α = lim sup γ,γ  Ω∗α xγ  yγ   ≤ ϕ η Ω∗α . Thus ϕ ⊗ η is bounded, so that X ∗∗ ⊗ Y ∗∗ ⊆ (X ∗ ⊗∗α Y ∗ )∗, and   (ϕ ⊗ η)Ω  ≤ ϕ η, ϕ ⊗ η∗∗α = sup Ω∗α Ω=0  

concluding the proof of (b).

Let X and Y be normed spaces and consider a function  · ∨ : X ⊗ Y → R  (sometimes denoted by ε(·)) defined for every  = i xi ⊗ yi ∈ X ⊗ Y as     ∨ = sup f (xi ) g(yi ),  i

f ≤1, g≤1, f ∈X ∗, g∈Y ∗

where the supremum is taken over all functionals f ∈ X ∗ with f  ≤ 1 and ∗ g ∈ Y with  (finite) representation i xi ⊗ yi of  ∈ X ⊗ Y.  g ≤ 1 for any ) = ⊗ g) does not depend on Note: (i) i f (xi ) g(y i (f ⊗ g)(xi ⊗ yi ) = (f i the representation i xi ⊗ yi of , and ∨ ≤ i xi yi . Now consider another function  · ∧ : X ⊗ Y → R (sometimes denoted by π(·)) defined by  ∧ = inf xi  yi  {xi }, {yi }, =Σi xi ⊗yi

i

for every  ∈ X ⊗ Y, where the infimum  is taken over all (finite) representations of  ∈ X ⊗ Y of the form  = i xi ⊗ yi . Theorem 7.4. Let X and Y be normed spaces. The functions  · ∨ and  · ∧ are reasonable crossnorms on X ⊗ Y. Moreover, a norm  · α is a reasonable crossnorm on X ⊗ Y if and only if ∨ ≤ α ≤ ∧

for every

 ∈ X ⊗ Y.

So  · ∨ is the least and  · ∧ is the greatest reasonable crossnorm on X ⊗ Y. Proof. Consider the functions  · ∨ : X ⊗ Y → R and  · ∧ : X ⊗ Y → R . (a)  · ∨ and  · ∧ are norms on X ⊗ Y.

140

7. Norms on Tensor Products

 If  = 0, then we may represent it as i xi ⊗ yi where each of {xi } and {yi } is linearly independent (cf. proof of Claim in the proof  of Corollary 4.10). Thus 0∨ = 0∧ = 0. By theabove mentioned claim, i f (xi )g(yi ) = 0 for 0 < f  and 0 < g implies i xi ⊗ yi = 0, and so ∨ = 0 implies  = 0. The proof that ∧ = 0 implies  = 0 is postponed until item (c), where it is shown that ∨ ≤ ∧. Moreover, it is clear that α∨ = |α| ∨ and α∧ = |α| ∧. Finally, we check the triangle inequality.      xi ⊗ yi + xj ⊗ yj   +  ∨ =  i j ∨      = sup f (xi ) g(yi ) + f (xj ) g(yj ) ≤ ∨ +  ∨.  f ≤1, g≤1

i

j

    For every ε > 0 there are representations  i xi ⊗ yi and j xj ⊗ yj of  and      such that i xi  yi  ≤ ∧ + ε and j xj  yj  ≤  ∧ + ε, and so      xi ⊗ yi + xi ⊗ yi   +  ∧ =  i j ∧    ≤ xi  yi  + xj  yj  ≤ ∧ +  ∧ + 2ε. i

j

(b)  · ∨ and  · ∧ are reasonable crossnorms on X ⊗ Y . Take arbitrary 0 = x ⊗ y ∈ X ⊗ Y and 0 = f ⊗ g ∈ X ∗ ⊗ Y ∗ . First note that x ⊗ y∨=

|f (x) g(y)| = sup |f (x)| sup |g(y)| = x y = x ⊗ y∧.

sup

f ≤1, g≤1

f ≤1

g≤1

∗ Now let  · ∗∨ be the norm  on (X ⊗∨ Y) when X ⊗ Y is equipped with  · ∨. Take an arbritrary  = i xi ⊗ yi ∈ X ⊗ Y. If f  = 0 and g = 0, then     f g |(f ⊗ g)()| = f  g  (x ) (y ) i g i  ≤ f  g ∨. f  i

∗

Thus f ⊗ g ∈ (X ⊗∨ Y) and f ⊗ g∗∨ ≤ f  g. So  · ∨ is a reasonable crossnorm according to Definition 7.1. Remark . Therefore if  · γ is any norm on X ⊗ Y (not necessarily a reasonable crossnorm), then  · ∨ ≤  · γ =⇒ X ∗ ⊗ Y ∗ ⊆ (X ⊗γ Y)∗ . ∗ Next let  · ∗∧ be the norm on (X ⊗∧ Y) when X ⊗ Y is equipped with  · ∧. Take an arbitrary ε > 0, and let i xi ⊗ yi be a representation of an arbitrary  in X ⊗ Y for which i xi  yi  ≤ ∧ + ε. Then      |(f ⊗ g)()| =  f (xi ) g(yi ) ≤ f  g xi  yi  ≤ f  g(∧ + ε). i

i

Thus, again, f ⊗ g ∈ (X ⊗∧ Y)∗ and f ⊗ g∗∧ ≤ f  g since the above inequality holds for every ε > 0. Hence  · ∧ is a reasonable crossnorm as well. (c) If  · α is a reasonable crossnorm on X ⊗ Y, then  · ∨ ≤  · α ≤  · ∧. Let   · α be a reasonable crossnorm on X ⊗ Y according to Definition 7.1. Let i xi ⊗ yi be any representation of an arbitrary  ∈ X ⊗ Y. Then

7.1 Reasonable Crossnorms

141

      α =  xi ⊗ yi  ≤ xi ⊗ yi α ≤ xi  yi  i

i

α

i



for all representations of . So α ≤ inf {xi },{yi } i xi  yi  = ∧. Also, letting  · ∗α be the norm on the dual (X ⊗ Y)∗ of X ⊗ Y when X ⊗ Y is equipped with the reasonable crossnorm  · α ,              f (xi )g(yi ) =  (f ⊗ g)(xi ⊗ yi ) = (f ⊗ g) xi ⊗ yi   i i i     ≤ f ⊗ g∗α  xi ⊗ yi  = f ⊗ g∗α α = f  g α . i

α

  So ∨ = supf ≤1,g≤1  i f (xi )g(yi ) ≤ α . Thus  · ∨ ≤  · α ≤  · ∧. (d) If a norm  · α on X ⊗ Y is such that  · ∨ ≤  · α ≤  · ∧, then it is a reasonable crossnorm. Consider Definition 7.1. For an arbitrary single tensor x ⊗ y in X ⊗ Y we get  · α ≤  · ∧

=⇒

x ⊗ yα ≤ x ⊗ y∧ = x y.

On the other hand, take an arbitrary single tensor f ⊗ g in X ∗ ⊗ Y ∗ . As we saw in item (b), |(f ⊗ g)()| ≤ f  g ∨ for every  ∈ X ⊗ Y. Therefore  · ∨ ≤  · α

=⇒

)| α f ⊗ g∗α = sup |(f⊗g)( ≤ sup f g ≤ f  g, α α =0

=0

where  · ∗α is the norm on the dual (X ⊗ Y,  · α )∗ of (X ⊗ Y,  · α ). Thus   x ⊗ yα ≤ x y and f ⊗ g ∈ (X ⊗ Y)∗ with f ⊗ g∗α ≤ f  g. The reasonable crossnorms  · ∨ and  · ∧ on X ⊗ Y are referred to as the injective norm and the projective norm, respectively. (Terminology will be justified in due course.) When equipped with a reasonable crossnorm  · α , a tensor product linear space X ⊗ Y yields different normed spaces: (X ⊗ Y,  · ∨), (X ⊗ Y,  · α ), and (X ⊗ Y,  · ∧). These will be denoted by X ⊗∨ Y, X ⊗α Y, and X ⊗∧ Y. Since  · ∨ ≤  · α ≤  · ∧, the reverse inclusion B∧ ⊆ Bα ⊆ B∨ holds, where B∨ = { ∈ X ⊗ Y : ∨ ≤ 1}, Bα = { ∈ X ⊗ Y : α ≤ 1}, B∧ = { ∈ X ⊗ Y : ∧ ≤ 1} are closed unit balls in X ⊗∨ Y, X ⊗α Y, X ⊗∧ Y. Even if X and Y are Banach spaces, a tensor product space X ⊗ Y equipped with an arbitrary reasonable crossnorm  · α , say X ⊗α Y = (X ⊗ Y,  · α ),  α Y. The may not be complete. The completion of X ⊗α Y is denoted by X ⊗ same notation  · α is used for the norms on both normed spaces X ⊗α Y and  α Y, both referred to as reasonable crossnorms. Saying that its completion X ⊗  X ⊗α Y is the completion of X ⊗α Y means X ⊗α Y is densely embedded in the  α Y. So X ⊗α Y is identified with the range R(J) = J(X ⊗α Y) Banach space X ⊗  α Y onto R(J) for which of an isometric isomorphism J : X ⊗α Y → R(J) ⊆ X ⊗ −   R(J) is dense in X ⊗α Y; that is, R(J) = X ⊗α Y (recall: the upper bar − stands for closure and ⊆ for densely included). If T ∈ B[X ⊗α Y, Z] for some  α Y, Z] be its extension over completion Banach space Z, then let T ∈ B[X ⊗ as in Theorem 5.1, which is represented by the diagram

142

7. Norms on Tensor Products

Z ⏐ T⏐

I

−−−→

Z = ⏐ −1 ⏐T |X ⊗  αY = T J

Z ⏐ ⏐T

J  α Y = J(X ⊗α Y)⊆X ⊗  α Y. X ⊗α Y −−−→ X ⊗

Since X ⊗α Y is identified with R(J) = J(X ⊗α Y), write X ⊗α Y for R(J), then n write  = i=1 xi ⊗ yi for elements in R(J), and finally, regarding X ⊗α Y as  α Y.  α Y, and by an abuse of notation, write (X ⊗α Y)− = X ⊗ being dense in X ⊗ Note that the linear space X ⊗ Y is already the result of an identification with some tensor product space T according to Corollary 3.6, and the further identification with R(J) remains (isometrically) identified with the original T . For normed spaces (in particular, for Banach spaces) X and Y, the comple Y (of X ⊗∨ Y and X ⊗∧ Y) are referred to as the injective  ∨Y and X ⊗ tions X ⊗ ∧ tensor product and the projective tensor product of X and Y, respectively. Remark 7.5. Regular Subspaces of Crossnormed Tensor Products. (a) Let S be any (algebraic) linear manifold of a linear space Z. Equip Z with a norm  · Z and consider the normed space (Z,  · Z ). Definition: S is a linear manifold of the normed space Z if, besides being an algebraic linear manifold of the linear space Z, it is equipped with the restriction to it of the norm on Z (same notation  · Z ), so that (S,  · Z ) is a normed space. In other words, if S inherits the norm from Z. (Recall: a norm defined on Z, when restricted to S, is a norm on the linear space S). Another way to say exactly the same thing (which will be useful in the sequel) is this. The identity I on S defines the trivial embedding S → Z (i.e., it defines the trivial linear-space isomorphism I : S → I(S) = S ⊆ Z). Suppose S is equipped with a norm  · S . We say that (S,  · S ) is a linear manifold of (Z,  · Z ) if and only if I(s)Z = sS for every s ∈ S. That is, I : (S,  · S ) → (Z,  · Z ) is an isometry ⇐⇒ S is a linear manifold of Z. (b) Let M and N be linear manifolds of normed spaces X and Y. Consider the regular linear manifold M ⊗ N of the linear space X ⊗ Y. Equip the linear spaces M ⊗ N and X ⊗ Y with a reasonable crossnorm  · α . Claim 1.

M ⊗α N may not be a linear manifold of X ⊗α Y.

Indeed, if a linear space X ⊗ Y is equipped with a reasonable crossnorm  · α , then it may happen that the restriction of  · α to a regular linear manifold M ⊗ N does not act on M ⊗ N as a norm inherited from X ⊗ Y, in the sense that M⊗α N = X ⊗α Y for some  ∈ M ⊗ N ⊆ X ⊗ Y, as discussed in (a). Example 1 .

M ⊗∧ N is not always a linear manifold of X ⊗∧ Y.

For instance, if a linear manifold M of a linear space X is properly included in X and if  = i ui ⊗ yi ∈ M ⊗ Y ⊂ X ⊗ Y, then it may occur that X ⊗∧ Y becomes less than M ⊗∧ Y . In fact,

7.1 Reasonable Crossnorms

M ⊗∧ Y = ≥



inf

ui ∈M, yi ∈Y, =Σi ui ⊗yi

inf



xi ∈X , xi ⊗yi =ui ⊗yi

i

i

143

ui ⊗ yi ∧

xi ⊗ yi ∧ = X

⊗∧ Y ,

 with each infimum taken over all representations of  = i ui ⊗ yi ∈ M ⊗ Y, while the infimum taken in the second expression runs over a larger set (including all ui ∈ M and also some possible xi ∈ X \ M). Then in general X

⊗∧ Y

≤ M ⊗∧ N ,

and the inequality may be strict (i.e., it may be that X ⊗∧ Y < M ⊗∧ N — e.g., by applying Theorem 7.9 below). Thus when the above inequality is strict for some  ∈ M ⊗ Y under the projective norm  · ∧, then M ⊗∧ Y is not a linear manifold of X ⊗∧ Y according to (a). Example 2 .

M ⊗∨ N is a linear manifold of X ⊗∨ Y.

This will be proved later in Theorem 7.19 (a little more familiarity with the injective norm  · ∨, as will be summarized in Remark 7.18, is needed). By the way, this is the reason why the injective norm is called injective. (c) Now let  · Z be the norm of a Banach space (Z,  · Z ) and let S be any linear manifold of the linear space Z. The identity I on S defines the trivial embedding S → Z. Instead of equipping S with the norm inherited from Z, equip it with an arbitrary norm  · S . Suppose (S,  · S ) is a Banach space and consider the trivial linear-space isomorphism I : S → I(S) = S ⊆ Z. Claim 2.

I : (S,  · S ) → (Z,  · Z ) is an isometry ⇐⇒ S is a subspace of Z.

Indeed, suppose I : (S,  · S ) → (I(S),  · Z ) is an isometric isomorphism. Since (S,  · S ) is complete and I(S) = S ⊆ Z, then (S,  · Z ) is complete (see, e.g., [52, Theorem 3.44]), and therefore the linear manifold S is closed in (Z,  · Z ), that is, S is a subspace of Z (see, e.g., [52, Corollary 3.41]). The converse comes from (a) since a subspace is a linear manifold. (d) Let X and Y be Banach spaces and suppose M and N are subspaces of X and Y (i.e., they are closed linear manifolds of X and Y inheriting the norms of X and Y), so that M and N are Banach spaces themselves. Consider the regular linear manifold M ⊗ N of the linear space X ⊗ Y. Equip them with an arbitrary reasonable crossnorm  · α. As we saw in (b), the crossnormed tensor product space M ⊗α N is not necessarily a linear manifold of the crossnormed  α N and X ⊗  α Y of tensor product space X ⊗α Y. Take the completions M ⊗ αN M ⊗α N and X ⊗α Y. If M ⊗α N is a linear manifold of X ⊗α Y, then M ⊗  α Y by Theorem 5.1(g). is a subspace of X ⊗ (e) According to (a), (c), (d), a way to show that M ⊗α N is a linear manifold  α N is a subspace of X ⊗  α Y, is to verify whether the of X ⊗α Y, and so M ⊗ trivial embeddings are isometries: for every  in M ⊗α N ,

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M ⊗α N = X

⊗α Y

=⇒ =⇒

M ⊗α N is a linear manifold of X ⊗α Y  α N is a subspace of X ⊗  α Y. M⊗

The converses hold trivially. (If S is any linear manifold of an arbitrary normed J J space Z and if Z −→J(Z) ⊆ Z and S −→J(S) ⊆ S for an arbitrary isometric sS =  sZ for every s ∈ S.) isomorphism J on Z, then sZ = sS =  We close the section with the notion of transpose norm  · t α . Let X and Y be linear spaces and consider the natural permutation isomorphism → Y ⊗ X between linear spaces X ⊗ Y and Y ⊗ X given  Π : X ⊗ Y by Π( i xi ⊗ yi ) = i yi ⊗ xi . For each  ∈ X ⊗ Y and each G ∈ Y ⊗ X set t

 = Π ∈ Y ⊗ X

and

t

G = Π −1 G ∈ X ⊗ Y.

This t  is the transpose of  and, symmetrically, t G is the transpose of G. So tt

 = t (t ) = Π −1 (t ) = Π −1 (Π ) = 

tt

G = t (t G) = Π(t G) = Π(Π −1 G) = G

for every for every

 ∈ X ⊗ Y,

G ∈ Y ⊗ X.

Now suppose X and Y are normed spaces. Let  · α be a reasonable crossnorm on Y ⊗ X . Associate with  · α a norm  · t α on X ⊗ Y defined as follows: t α = t α

for every

 ∈ X ⊗ Y for which t  ∈ Y ⊗α X .

Consider the iterated norm  · tt α =  · t (t α) on Y ⊗ X defined by Gt t α = t Gt α Since

tt

for every

G ∈ Y ⊗ X for which t G ∈ X ⊗ Y.

G = G, and by the above two identities, we get Gα = tt Gα = t Gt α = Gtt α

for every

G ∈ Y ⊗ X,

and hence X ⊗tt α Y = X ⊗α Y because  · t t α =  · α . Theorem 7.6. If  · α is a reasonable crossnorm on Y ⊗ X , then (a)  · t α is a reasonable crossnorm on X ⊗ Y,

and

(b) the natural permutation Π : X ⊗t α Y → Y ⊗α X is an isometric isomorphism so that X ⊗t α Y ∼ = Y ⊗α X . If  · β is another reasonable crossnorm on Y ⊗ X , then (c)  · α ≤  · β if and only if  · t α ≤  · t β . Proof. (a) Let  · α be a reasonable crossnorm acting on the tensor product space of an arbitrary pair of normed spaces, say,  · α : Y ⊗ X → R . It is clear that  · t α : X ⊗ Y → R defines a norm acting on X ⊗ Y for which x ⊗ yt α = Π(x ⊗ y)α = y ⊗ xα = y x

7.1 Reasonable Crossnorms

145

for every single tensor x ⊗ y in X ⊗ Y. Moreover, take an arbitrary single tensor f ⊗ g in X ∗ ⊗ Y ∗ ⊆ X  ⊗ Y  ⊆ (X ⊗ Y) . Recall that (X ⊗t α Y)∗ = {Λ ∈ (X ⊗ Y) : supt α ≤1, ∈X ⊗Y |Λ()| < ∞}, and also that for any  =    i xi ⊗ yi = i f (xi )g(yi ) =  (f ⊗ g) = (f ⊗ g) i xi ⊗ yi ∈ X ⊗ Y we get t g(y )f (x ) = (g ⊗ f ) y ⊗ x = (g ⊗ f )  . Therefore i i i i i i f ⊗ g(X ⊗t α Y)∗ = =

sup t α ≤1, ∈X ⊗Y

sup Gα ≤1, G∈Y⊗X

|(f ⊗ g)| =

sup t α ≤1, t ∈Y⊗X

|(g ⊗ f ) t |

|(g ⊗ f )G| = g ⊗ f (Y ⊗α X )∗ = gf ,

so that f ⊗ g lies in (X ⊗t α Y)∗ and f ⊗ g∗t α = f ⊗ g(X ⊗t α Y)∗ = f g, and hence  · t α is a reasonable crossnorm on X ⊗ Y by Theorem 7.2. (b) Since Π Y ⊗α X = t Y ⊗α X = X ⊗t α Y for every  ∈ X ⊗t α Y, Π : X ⊗t α Y → Y ⊗α X is an isometric isomorphism. (c) Thus t α = Π α ≤ Π β = t β for every  ∈ X ⊗ Y whenever   · α ≤  · β . For the converse, set  · α =  · t t α and  · β =  · tt β .  This norm  · t α : X ⊗ Y → R is the transpose norm of  · α : Y ⊗ X → R . Remark 7.7. Symmetric Reasonable Crossnorms. A reasonable crossnorm is defined for a given pair of normed spaces X and Y. However, suppose for a moment that a reasonable crossnorm  · α may equip both linear spaces X ⊗ Y and Y ⊗ X (as will be the case for the uniform crossnorms of Chapter 8). So its transpose  · t α also equips both linear spaces Y ⊗ X and X ⊗ Y. Define: (a)  · α is a symmetric reasonable crossnorm if  · t α =  · α . ∼ Y ⊗α X , which implies By Theorem 7.6(b), if  · t α =  · α , then X ⊗α Y = Π : X ⊗α Y → Y ⊗α X is an isometric isomorphism. Therefore ∼ Y ⊗α X (b)  · α is a symmetric reasonable crossnorm ⇐⇒ X ⊗α Y =

Π : X ⊗α Y → Y ⊗α X is an isometric isomorphism.  α Y and Y ⊗  α X be completions of X ⊗α Y and Y ⊗α X , Moreover, let X ⊗    and let Π in B[X ⊗α Y, Y ⊗α X ] be the extension over completion of Π in B[X ⊗α Y, Y ⊗α X ], which are isometric isomorphisms together (see Theorem 5.1(d)). Hence Π : X ⊗α Y → Y ⊗α X is an isometric isomorphism if and only  α X , which means the norms  · α (same notation) on X ⊗α Y αY ∼ if X ⊗ =Y⊗  and on X ⊗α Y are symmetric reasonable crossnorms together. Summing up, ⇐⇒

(c)  · α is a symmetric reasonable crossnorm (d) Furthermore,

⇐⇒

αY ∼ αX . X⊗ =Y⊗

the projective and the injective norms are symmetric.

Indeed,  · ∧ and  · ∨ can be assigned to X ⊗ Y and to Y ⊗X . Take  =  x ⊗ y i i in X ⊗ Y. (i) If  lies in X ⊗∧ Y, then ∧ = inf i i xi  yi  = inf i yi xi = t ∧ = t ∧. Also, (ii) if lies in X ⊗∨ Y, then ∨ = supf =g=1 | i f (xi )g(yi )| = supg=f =1 | i g(yi )f (xi )| = t ∨ = t ∨.

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7. Norms on Tensor Products

7.2 Projective Tensor Product Take a tensor product space X ⊗ Y of a pair of normed spaces X and Y. Consider the characterization of a reasonable crossnorm  · α as in Theorem 7.2, which shows that  x ⊗ yα = x y for every single tensor x ⊗ y ∈ X ⊗ Y. Let the finite sum i xi ⊗ yi be any representation of an arbitrary  ∈ X ⊗ Y. If  · α is a reasonable crossnorm on the linear space X ⊗ Y, then   α ≤ xi ⊗ yi α = xi  yi  i

i

for every representation of . The projective norm ∧ is precisely the infimum of the above sum taken over all possible representations of . Let M and N be linear manifolds of normed spaces X and Y, respectively. Take the regular linear manifold M ⊗ N of the linear space X ⊗ Y. The projective norm can be assigned to both spaces M ⊗ N and X ⊗ Y. Thus consider the normed spaces M ⊗∧ N = (M ⊗ N ,  · ∧) and X ⊗∧ Y = (X ⊗ Y,  · ∧). If X and Y are Banach spaces and M and N are subspaces of them, then  Y. In light of the previous Remark  N and X ⊗ take their completions M ⊗ ∧ ∧ 7.5(b) on linear manifolds and subspaces of tensor products, we begin by establishing when M ⊗∧ N is a linear manifold of the normed space X ⊗∧ Y  Y.  N is a subspace of the Banach space X ⊗ and, consequently, when M ⊗ ∧ ∧ Theorem 7.8. Suppose M and N are complemented subspaces of Banach spaces X and Y, respectively. (a) Then the projective norm on M ⊗∧ N and the norm on M ⊗ N inherited from X ⊗∧ Y are equivalent. (b) If M ⊗∧ N is a linear manifold of the normed space X ⊗∧ Y, then it is closed in X ⊗∧ Y (thus a subspace of X ⊗∧ Y). (c) If M ⊗∧ N is a linear manifold of the normed space X ⊗∧ Y, then M ⊗∧ N  N is is a complemented subspace of the normed space X ⊗∧ Y, and M ⊗ ∧  a complemented subspace of the Banach space X ⊗∧ Y. (d) If M = R(E) and N = R(P ) for projections E in B[X ] and P in B[Y] with E = P  = 1, then M ⊗∧ N is a linear manifold of X ⊗∧ Y. Proof. Suppose M and N are complemented in X and Y. Let E in B[X ] and P in B[Y] be projections such that M = R(E) and N = R(P ) (Remark 5.3). Take the tensor product transformation E ⊗ P in L[X ⊗ Y] (Theorem 3.17), which is idempotent, thus a projection on the linear space X ⊗ Y. Indeed,   Exi ⊗ P yi = E 2 xi ⊗ P 2 yi = (E ⊗ P ) (E ⊗ P )(E ⊗ P ) = (E ⊗ P ) for every  =

i

 i

i

xi ⊗ yi ∈ X ⊗Y. Also (recall Theorem 3.19(e)) R(E ⊗ P ) = M ⊗ N ,

7.2 Projective Tensor Product

147

so that  = (E ⊗ P ) for every  in the regular linear manifold M ⊗ N of the linear space X ⊗ Y. Equip X ⊗ Y with the projective norm and take  ∈ X ⊗∧ Y = (X ⊗Y,  · ∧) = (X ⊗Y,  · X ⊗∧Y ). Since E and P are bounded, sup

0=∈X ⊗∧Y

(E⊗P )X ⊗∧Y X ⊗∧Y

=

sup

0=∈X ⊗∧Y

 inf {xi }, {yi } i Exi P yi   inf {xi }, {yi } i xi yi 

≤ E P .

The above shows that the linear transformation E ⊗ P as considered in Theorem 3.17 is bounded when acting from the normed space X ⊗∧ Y to itself. (We will consider bounded linear tensor product transformations in detail in due course.) Thus E ⊗ P is a continuous projection, E ⊗ P ∈ B[X ⊗∧ Y], with E ⊗ P B[X ⊗∧Y] ≤ E P . On the other hand, since  · X ⊗∧Y is a reasonable crossnorm, ExP y = Ex⊗P yX ⊗∧Y = (E ⊗P )(x⊗y)X ⊗∧Y ≤ E ⊗P B[X ⊗∧Y] xy for every x ∈ X and y ∈ Y. Hence P y E P  = sup Ex x sup y ≤ E ⊗ P B[X ⊗∧Y] . x=0

y=0

Since E and P are continuous projections, 1 ≤ E and 1 ≤ P , and so 1 ≤ E ⊗ P B[X ⊗∧Y] = E P . (a) Takean arbitrary  ∈ M ⊗ N . Consider a representation of it in X ⊗ Y, say  = i xi ⊗ yi . Recall from the argument in Remark 7.5(b) that this representation  may occurfor xi ∈ X \M and yi ∈ Y\N . Since  = (E ⊗ P ), M ⊗ N with (E ⊗ P ) i xi ⊗ yi = i Exi ⊗ P yi is a representation of  in  Exi ∈ M and P xi ∈ N. Since M ⊗∧ N = inf {ui },{vi }, =Σi ui ⊗vi i ui  vi ,   M⊗∧N ≤ Exi  P yi  ≤ E P  xi  yi , i

i

and hence M⊗∧N ≤

inf

{xi }, {yi }, =Σi xi ⊗yi

 EP  xi yi  = EP X ⊗∧Y . i

Therefore, with the help of Remark 7.5(b), X ⊗∧Y ≤ M⊗∧N ≤ E P  X ⊗∧Y ,

(∗)

and so the normed spaces M ⊗∧ N = (M ⊗ N ,  · ∧) = (M ⊗ N ,  · M⊗∧N ) and (M ⊗ N ,  · X ⊗∧Y ) with a common underlying linear space M ⊗ N are equipped with equivalent norms, which proves (a). (b) The range of the above continuous projection E ⊗ P in B[X ⊗∧ Y] is given by R(E ⊗ P ) = (M ⊗ N ,  · X ⊗∧ Y ), which is a linear manifold of the normed

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7. Norms on Tensor Products

space X ⊗∧ Y inheriting the norm of X ⊗∧ Y. But ranges of continuous projections are closed. Thus this is a closed linear manifold of X ⊗∧ Y, and hence a subspace of X ⊗∧ Y. Now if M ⊗∧ N is a linear manifold of X ⊗∧ Y, then it coincides with (M ⊗ N ,  · X ⊗∧ Y ) and so it is a subspace of X ⊗∧ Y. (c) Suppose M ⊗∧ N is a linear manifold of the normed space X ⊗∧ Y, so that it is a subspace of X ⊗∧ Y according to (b). Let E ⊗ P in B[X ⊗∧ Y] be the projection with R(E ⊗ P ) = M ⊗∧ N , as in the proof of (b) above. Thus M ⊗∧ N  in B[X ⊗  Y] be the extension is complemented by Remark 5.3. Now let E ⊗P ∧ over completion of E ⊗ P in B[X ⊗∧ Y]. This is the extension by continuity of J(E ⊗ P )J −1 for some isometric isomorphism J, so that J(E ⊗ P )J −1 is a continuous projection with R(J(E ⊗ P )J −1 ) = J(R(E ⊗ P )) = J(M ⊗∧ N ), and  Y of  P into the closure J(X ⊗∧ Y)− = X ⊗ so is its extension by continuity E ⊗ ∧  is a continuous projection with the same closed range J(X ⊗∧ Y). Thus E ⊗P  N. (See  P ) = R(J(E ⊗ P )J −1 ) = J(R(E ⊗ P )) = J(M ⊗∧ N ) = M ⊗ R(E ⊗ ∧ ∧  N Theorem 5.1(a,f) and the diagram in the proof of Theorem 5.1.) So M ⊗ ∧  is a complemented subspace of X ⊗∧ Y (see Remark 5.3 once again). (d) If the complemented M and N are such that M = R(E) and N = R(P ) for projections E ∈ B[X ] and P ∈ B[Y] with E = P  = 1, then by (∗) M ⊗∧ N = X

⊗∧ Y .

Therefore, according to Remark 7.5(e), this means M ⊗∧ N is a linear manifold   of the normed space X ⊗∧ Y, which proves (d).  Y is the completion of the normed Since a projective tensor product X ⊗ ∧ space X ⊗∧ Y = (X ⊗ Y,  · ∧), this X ⊗∧ Y is naturally identified with the  range J(X ⊗∧Y) of an isometric nisomorphism J : X ⊗∧Y → J(X n⊗∧ Y) ⊆ X ⊗∧Y, as we saw above. Write  = i=1 xi ⊗ yi with ∧ = inf i=1 xi  yi  for the elements in J(X ⊗∧ Y) identified with X ⊗∧ Y, regard X ⊗∧ Y as being  Y \X ⊗∧ Y for X ⊗  Y \J(X ⊗∧ Y).  Y, and write X ⊗ dense in X ⊗ ∧ ∧ ∧ The next result is sometimes referred to as the Grothendieck Theorem. In fact, most of the results in this chapter are Grothendieck’s (see, e.g., [17]). Theorem 7.9. (Grothendieck). If X and Y are Banach spaces, then for  Y there exist X -valued and Y-valued sequences {xk } and {yk } every  ∈ X ⊗ ∧ for which the real sequence {xk  yk } is summable and  = xk ⊗ yk k

 (i.e.,  every  ∈ X ⊗∧ Y is representable in the form of a countable sum  = x ⊗ y in the sense that it is either a finite or a countably infinite reprek k k  Y is given by sentation). Moreover, the norm  · ∧ on X ⊗ ∧  xk  yk , ∧ = inf k   Y. where the infimum is taken over all representations k xk ⊗ yk of  ∈ X ⊗ ∧

7.2 Projective Tensor Product

149

 Y be the completion of X ⊗∧ Y. Take any  ∈ X ⊗  Y \X ⊗∧ Y Proof. Let X ⊗ ∧ ∧ (otherwise the resulting finite sum is trivially obtained). Thus  is arbitrarily close to elements nk in X ⊗∧ Y. Take an arbitrary ε > 0. For each positive integer xi ⊗ yi in X ⊗∧ Y such that k take k = i=1  − k ∧
1 n1 k nj+1 nk+1 xi ⊗ yi ∧ = xi  yi  + xi  yi  i=1

i=1

< ∧ + ε + ε

k

j=1

i=nj +1

1 j j=1 2

< ∧ + 2ε.  Hence the sequence {xi ⊗ yi } is absolutely summable with i xi ⊗ yi ∧ <  Y is a Banach space. ∧ + 2ε. Then the sequence is summable because X ⊗ ∧ nk  Y. This means the sequence of partial sums { i=1 xi ⊗ yi } converges in X ⊗ ∧ nk  Y of k = i=1 xi ⊗ yi ∈ X ⊗∧ Y is represented as Thus the limit  ∈ X ⊗ ∧  = xi ⊗ yi , i and   xi ⊗ yi ∧ = xi  yi  ≤ ∧ + 2ε, ∧ ≤ i i  of  is the infimum of i xi  yi  over all so that the projective norm ∧   representations of  of the form i xi ⊗ yi . This is the classical statement of Theorem 7.9. Do we need that X and Y be complete? And X and Z in the next theorem? And X and Y in its corollary? As for the next result, recall that (i) by a subspace of a normed space we mean a closed linear manifold of it, (ii) a normed space Y is a quotient of a normed space X if there is a quotient transformation Q of X onto Y, (iii) the natural quotient map π of X onto the quotient space X/M is a quotient transformation, and so X/M is a quotient of X, for every proper subspace M of X .

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7. Norms on Tensor Products

The next theorem justifies the terminology projective for the norm  · ∧. It shows that when tensor products are equipped with the projective norm, then the fact that X /M is a quotient of X is transferred (projected) to tensor  X for every Z.  (X /M) is a quotient of Z ⊗ products in the sense that Z ⊗ ∧ ∧  (X /M) Theorem 7.10. If M is a subspace of a Banach space X , then Z ⊗ ∧  X for every Banach space Z. is a quotient of Z ⊗ ∧ Proof. Suppose all linear spaces are nonzero. Let M be a subspace of a Banach space X and take an arbitrary Banach space Z. Consider the tensor product  (X /M). Take an arbitrary G ∈ Z ⊗  (X /M). Since Z  X and Z ⊗ spaces Z ⊗ ∧ ∧ ∧ and X /M are Banach spaces, we may apply Theorem 7.9, and so there are Z-valued and X /M-valued sequences {zk } and {[xk ]} such that     zk ⊗ [xk ] with GZ ⊗∧(X /M) = inf zk Z [xk ] X /M . G= k

k

Let π : X → X /M be the natural quotient map, which is a linear surjective contraction (see Sections 1.3 and 5.3), and consider the transformation Q =  (X /M) where I is the identity on Z, which is linear (see  X → Z⊗ I ⊗ π: Z⊗ ∧ ∧   X for any X Section 3.3). It is surjective. In fact, if  = k zk ⊗ xk ∈ Z ⊗ ∧ valued sequence {xk } with xk ∈ [xk ], then Q( ) = G (Definition 3.16). Also, Q  is a contraction. Indeed, an arbitrary  ∈ Z ⊗∧ X can be represented as  = y ⊗ v for some Z-valued sequence {y k k } and some X -valued sequence k k   {vk } by Theorem 7.9, so that (as [v] X /M ≤ v  X for every v  ∈ [v] ⊆ X )     Q()Z ⊗∧(X /M)= inf yk Z [vk ] X /M ≤ inf yk Z vk X = Z ⊗∧X . k

k

Now take an arbitrary ε > 0. Associated with ε there is a Z-valued sequence sequence {z  {[xk ]} such that G =   k } with zk Z ≤ 1 and an X /M-valued   [x z ⊗ [x ] with G = inf z  ]  k Z ⊗ (X /M) k X /M and k k k k Z ∧ 

  zk Z [xk ] X /M ≤ GZ ⊗∧(X /M) + ε 12 . (∗) k   Since [xk ] = xk + M and [xk ] X /M = inf u∈M xk + uX ≤ xk X , then for each k there exists xk ∈ [xk ] (so that [xk ] = [xk ]) for which     1 [xk ]  ≤ xk X ≤ [xk ] X /M + ε 2k+1 . (∗∗) X /M     X we get G = k zk ⊗[xk ] = k zk ⊗[xk ] = Thus with  = k zk ⊗xk ∈ Z ⊗ ∧ Q( ). Hence by (∗) and (∗∗), and recalling that zk Z ≤ 1, GZ ⊗∧(X /M) ≤

 GZ ⊗∧(X /M) = Q( )Z ⊗∧(X /M) ≤  Z ⊗∧X = inf zk Z xk X k      1 ≤ zk Z xk X ≤ zk Z [xk ] X /M + ε 2k+1 k  k   1   zk Z [xk ] +ε ≤ 2k+1 k

X /M

k

≤ GZ ⊗∧(X /M) + ε 12 + ε 12 = GZ ⊗∧(X /M) + ε.

7.2 Projective Tensor Product

151

 (X /M) there is an  = As this holds for every ε > 0, then for every G ∈ Z ⊗ ∧ −1  Q (G) ∈ Z ⊗∧ X such that GZ ⊗∧(X /M) = Z ⊗∧X . So the linear surjective  X →Z⊗  (X /M) is a quotient transformation (Theorem contraction Q : Z ⊗ ∧ ∧  X (Definition 5.12).  (X /M) is a quotient of Z ⊗   5.13(a)). Thus Z ⊗ ∧ ∧ Corollary 7.11. If M and N are subspaces of Banach spaces X and Y, then  Y.  (Y/N ) is a quotient of X ⊗ (X /M) ⊗ ∧ ∧  Z  Z is a quotient of X ⊗ Proof. Symmetrically (by Theorem 7.10), (X /M) ⊗ ∧ ∧ for every Banach space Z. Since a quotient space of a Banach space modulo a subspace is a Banach space (cf. Remark 5.9), and since composition of quotient transformations is again a quotient transformation (according to Theorem  Y.  (Y/N ) is a quotient of X ⊗   5.13(d)), then by transitivity (X /M) ⊗ ∧ ∧ Theorem 3.7 played a central role in the theory of tensor products of linear spaces. Theorem 7.12 below is a counterpart of it, which holds when the tensor product space is equipped with a suitable reasonable crossnorm. This plays an equally crucial role in the theory of tensor products of Banach spaces, and the suitable reasonable crossnorm is precisely the projective norm. The next result is often referred to as the universal mapping property of the projective norm, or the universal mapping principle of the projective tensor product. It says that a bilinear map φ : X ×Y → Z is continuous if and only if its linearization Φ : X ⊗∧ Y → Z is. In its classical form it reads as follows. Theorem 7.12. Take an arbitrary triple (X , Y, Z) of Banach spaces. The  Y, Z] are isometrically isomorphic: Banach spaces b[X ×Y, Z] and B[X ⊗ ∧  Y, Z]. b[X ×Y, Z] ∼ = B[X ⊗ ∧ Proof. Take the tensor product space X ⊗ Y and the natural bilinear map θ ∈ b[X ×Y, X ⊗ Y] associated with it. Theorem 3.7 ensures the existence of a linear-space isomorphism Cθ : L[X ⊗ Y, Z] → b[X ×Y, Z] between the linear spaces L[X ⊗ Y, Z] and b[X ×Y, Z] defined as the composition with θ: Cθ (Φ) = Φ ◦ θ ∈ b[X ×Y, Z]

for every

Φ ∈ L[X ⊗ Y, Z].

Let X and Y be Banach spaces and equip the linear space X ⊗ Y with the projective norm to get the normed space X ⊗∧ Y. Suppose Z is a Banach space and consider the Banach space B[X ⊗∧ Y, Z], which is a linear manifold of L[X ⊗∧ Y, Z]. Let J be the restriction to B[X ⊗∧ Y, Z] of the linear-space isomorphism Cθ on L[X ⊗∧ Y, Z], which remains linear and injective, J = Cθ |B[X

⊗∧ Y,Z] :

B[X ⊗∧ Y, Z] → R(J ) ⊆ b[X ×Y, Z].

We show that (a) R(J ) = b[X ×Y, Z] and (b) J is an isometry onto R(J ). Let  · B and  · b stand for the norms in B[X ⊗∧ Y, Z] and b[X ×Y, Z].

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7. Norms on Tensor Products

(a1 ) If Φ ∈ B[X ⊗∧ Y, Z], then J (Φ) = Cθ (Φ) = Φ ◦ θ ∈ b[X ×Y, Z] is such that φ(x, y) = J (Φ)(x, y) = (Φ ◦ θ)(x, y) = Φ(x ⊗ y)

for every

(x, y) ∈ X ×Y,

and hence J (Φ)(x, y)Z = Φ(x ⊗ y)Z ≤ ΦB x ⊗ y∧ = ΦB xX yY .

(∗)

Then the bilinear map J (Φ) is bounded. Thus J (Φ) ∈ b[X ×Y, Z] for every Φ in B[X ⊗∧ Y, Z], and so R(J ) ⊆ b[X ×Y, Z]. (a2 ) Conversely, if φ ∈ b[X ×Y, Z], then there is a unique Φ ∈ B[X ⊗∧ Y, Z] such that Cθ (Φ) = Φ ◦ θ = φ. So Φ(x ⊗ y)Z = Φ ◦ θ(x, y)Z = φ(x, y)Z ≤ φb xX yY  for every (x, y) ∈ X ×Y. Hence for an arbitrary  = i xi ⊗ yi ∈ X ⊗∧ Y,          xi ⊗ yi  =  Φ(xi ⊗ yi ) Φ()Z = Φ i i Z Z   ≤ φb xi X yi Y = φb xi X yi Y . i

i

 As this holds  for every (finite) representation of  = i xi ⊗ yi , and since ∧ = inf i xi X yi Y over all representations, then for every  ∈ X ⊗∧ Y Φ()Z ≤ φb ∧,

(∗∗)

and the linear Φ is bounded: Φ ∈ B[X ⊗∧ Y, Z]. So φ = Cθ (Φ) = J (Φ) ∈ R(J ). Since this holds for every φ ∈ b[X ×Y, Z], we get b[X ×Y, Z] ⊆ R(J ). (a) By (a1 ) and (a2 ) we get R(J ) = b[X ×Y, Z]. (b) Take an arbitrary Φ ∈ B[X ⊗∧ Y, Z]. As we saw above in (∗), J (Φ)(x, y)Z ≤ ΦB xX yY for every (x, y) ∈ X ×Y. Then the bilinear map J (Φ) is bounded with norm J (Φ)b ≤ ΦB . Conversely, the unique φ = Cθ (Φ) = J (Φ) is such that φ ∈ b[X ×Y, Z] because R(J ) ⊆ b[X ×Y, Z] by (a1 ) and hence, as we saw above in (∗∗), Φ()Z ≤ φb ∧ = J (Φ)b ∧ for every  ∈ X ⊗∧ Y. Therefore, ΦB ≤ φb = J (Φ)b . Thus J (Φ)b = ΦB for every Φ ∈ B[X ⊗∧ Y, Z]. That is, J is an isometry.

7.2 Projective Tensor Product

153

(c) Then the linear transformation J : B[X ⊗∧ Y, Z] → b[X ×Y, Z] is a surjective isometry by (a) and (b), which means J is an isometric isomorphism. So B[X ⊗∧ Y, Z] ∼ = b[X ×Y, Z].  Y of X ⊗∧ Y. According to Theorem 5.1(e), Consider the completion X ⊗ ∧  Y, Z] ∼ B[X ⊗ = B[X ⊗∧ Y, Z] ∧ because Z is a Banach space. So the stated result follows by transitivity.

 

Do we need that X and Y in Theorem 7.12 be Banach spaces? Remark 7.13. Dualities and the Projective Universal Mapping Principle. (a) Let X , Y, Z be arbitrary linear spaces (over the same field F ) and let ∼ = denote linear-space isomorphism. From a purely algebraic point of view, Theorem 2.6, Remark 2.7(b) and Theorem 3.7 ensure that L[X ⊗ Y, Z] ∼ = b[X ×Y, Z] ∼ = L[X , L[Y, Z]] ∼ = L[Y, L[X , Z]]. (b) In particular, if Z = F and  stands for algebraic dual, and recalling Remarks 3.8(b) and 2.7(b) again, X  ⊗ Y  ⊆ (X ⊗ Y) ∼ = L[X , Y  ] ∼ = L[Y, X  ]. = b[X ×Y, F ] ∼ (c) If X , Y, Z are Banach spaces and X ⊗ Y is equipped with the projective  Y, Z] is naturally isometrically isomorphic to B[X ⊗∧ Y, Z], norm, then B[X ⊗ ∧ which in turn is isometrically isomorphic to b[X ×Y, Z] by Theorem 7.12. Thus using Theorem 6.3 (with the help of Remark 6.4(a)), where ∼ = now means isometric isomorphism, we also get  Y, Z] ∼ B[X ⊗ = B[X ⊗∧ Y, Z] ∼ = b[X ×Y, Z] ∼ = B[X , B[Y, Z]] ∼ = B[Y, B[X , Z]]. ∧ (d) Now set Z = F and replace  by the normed-space dual ∗ . By Definition 7.1, X ∗ ⊗ Y ∗ ⊆ (X ⊗α Y)∗ for any reasonable crossnorm  · α on X ⊗ Y. By Theorem 7.3, the induced uniform norm  · ∗α on (X ⊗α Y)∗ is a reasonable crossnorm on X ∗ ⊗ Y ∗ when restricted to it. Equip it with this norm and get X ∗ ⊗∗α Y ∗ . Thus X ∗ ⊗∗α Y ∗ ⊆ (X ⊗α Y)∗ . (Note: X ∗ ⊗∗α Y ∗ is a linear manifold of the normed space (X ⊗α Y)∗ — its norm is inherited from (X ⊗α Y)∗ .) Thus  ∗α Y ∗ ⊆ (X ⊗α Y)∗ , which is its own natural Theorem 5.1(g) ensures that X ∗ ⊗ completion since dual spaces are complete. Moreover, Theorem 5.1(f) says  α Y)∗ and X ∗ ⊗∗α Y ∗ ∼  ∗α Y ∗ . Hence that (X ⊗α Y)∗ ∼ = (X ⊗ = X∗ ⊗  α Y)∗  ∗α Y ∗ ∼ X∗ ⊗ = X ∗ ⊗∗α Y ∗ ⊆ (X ⊗α Y)∗ ∼ = (X ⊗ for every reasonable crossnorm  · α on X ⊗ Y. Also, by Remark 6.4(a), b[X ×Y, F ] ∼ = B[X , Y ∗ ] ∼ = B[Y, X ∗ ].

154

7. Norms on Tensor Products

 Y)∗ ∼ (e) By setting Z = F in Theorem 7.12 we get (X ⊗ = b[X ×Y, F ]. Thus ∧ ∗ ∗ ∗ ∗ ∗  ∗∧Y ∗ ∼  Y)∗ ∼ X∗ ⊗ = B[X , Y ] ∼ = B[Y, X ] = X ⊗∗∧ Y ⊆ (X ⊗∧ Y) ∼ = (X ⊗ = b[X ×Y, F ] ∼ ∧

for the projective norm  · ∧ (compare with Remark 4.9(e,f,g,h)). It is worth noticing that although the above equivalences (i.e., isometric isomorphisms)  Y)∗ of the projective tensor product, give interpretations for the dual (X ⊗ ∧ the above inclusion (embedding) does not supply any interpretation for the  Y itself (but it gives interpretations for the projective tensor product X ⊗ ∧ ∗  ∗ tensor product X ⊗ ∗∧Y under the reasonable crossnorm  · ∗∧).  Y)∗ ∼ (f) The proof of Theorem 7.12 for Z = F shows that (X ⊗∧ Y)∗ ∼ = (X ⊗ = ∧ b[X ×Y, F ], where an arbitrary isometrically isomorphic pair Λ ∈ (X ⊗∧ Y)∗ ∼ = φ ∈ b[X ×Y, F ] is related by Λ(x ⊗ y) = φ(x, y) for every (x, y) ∈ X ×Y. Then    Λ() = Λ(xi ⊗ yi ) = φ(xi , yi ) for every  = xi ⊗ yi ∈ X ⊗∧ Y, i

i

i

as Λ is linear. (Note: the value Λ() of the functional Λ at a point  does not depend on the representation of  — cf. Remark 3.18(a).) As Λ is bounded and ∼ Λ∗∧ = φ. This yields another expres= stands for isometric isomorphism, sion for the projective norm of  = i xi ⊗ yi ∈ X ⊗∧ Y (cf. Remark 5.2(d)),     sup |Λ()| = sup φ(xi , yi ). ∧ =  Λ∗∧≤1, Λ∈(X ⊗∧ Y)∗

φ≤1, φ∈b[X ×Y, F ]

i

(g) On the other hand, by Definition 7.1 the linear space X ∗ ⊗ Y ∗ is a linear manifold of the linear space (X ⊗α Y)∗ = B[X ⊗α Y, F ], which is isometrically  α Y)∗ (by Theorem 5.1(e)) for any reason α Y, F ] = (X ⊗ isomorphic to B[X ⊗ able crossnorm. Then the linear space X ∗ ⊗∧ Y ∗ (now equipped with projective norm) remains algebraically embedded in the Banach space (X ⊗∧ Y)∗, X ∗ ⊗∧ Y ∗ ⊆ (X ⊗∧ Y)∗ ,  Y)∗ ,  Y ∗ remains algebraically embedded in (X ⊗ and so its completion X ∗ ⊗ ∧ ∧  Y)∗ ,  Y ∗ ⊆ (X ⊗ X∗⊗ ∧ ∧ where, differently from items (d) and (e) above, ⊆ now means linear-space embedding but not normed-space embedding as the norms are different:  · ∧  Y)∗ = B[X ⊗  Y, F ]. Next take an arbi Y ∗ and  · ∗∧ equips (X ⊗ equips X ∗ ⊗ ∧ ∧ ∧ ∗ ∗ ∗  that there are X ∗-valued trary Ω ∈ X ⊗∧ Y ⊆ (X ⊗∧ Y) . Theorem 7.9 ensures  ∗ and Y -valued sequences {fj } and {gj } with j fj gj  < ∞ such that   Ω= fj ⊗ gj and Ω∧ = inf fj gj , j

j



where the infimum is taken over all representations j fj ⊗ gj of Ω. Hence   

       fj ⊗ gj xi ⊗ yi  =  fj (xi ) gj (yi ) |Ω()| =  j i j i 



  ≤ fj gj xi yi  ≤ fj gj  xi yi  j

i

j

i

7.2 Projective Tensor Product

155

   Y with ∧ = inf i xi yi  (Theorem 7.9). for every  = i xi ⊗ yi ∈ X ⊗ ∧ So |Ω()| ≤ Ω∧∧. Then Ω∗∧ = sup∧≤1 |Ω()| ≤ Ω∧. Thus Ω∗∧ ≤ Ω∧

for every

 Y)∗ .  Y ∗ ⊆ (X ⊗ Ω ∈ X∗⊗ ∧ ∧

(h) Next take the isometric isomorphism J : (X ⊗∧ Y)∗ → b[X ×Y, F ] in the proof of Theorem 7.12 with Z = F , where the value of J at an arbitrary single tensor f ⊗ g ∈ X ∗ ⊗∧ Y ∗ ⊆ (X ⊗∧ Y)∗ is such that J (f ⊗ g) = φ ∈ b[X ×Y, F ] ∗

∗

if and only if

φ(· , ·) = f (·)g(·).

∗

Indeed, if f ⊗ g lies in X ⊗ Y ⊆ (X ⊗∧ Y) = B[X ⊗∧ Y, F ], then   J (f ⊗g)(x, y) = (f ⊗g)◦θ (x, y) = (f ⊗g)θ(x, y) = (f ⊗g)(x⊗y) = f (x)g(y) for every (x, y) ∈ X ×Y (cf. proof of Theorem 3.7 with Cθ = J and Φ = f ⊗ g, and see also Definition 3.16). Conversely, if φ(· , ·) = f (·)g(·)  lies in b[X ×Y, F ], then J −1 (φ) = f ⊗ g. Therefore, for an arbitrary Ω = j fj ⊗ gj ∈ X ∗ ⊗ Y ∗,  J (Ω) = φ ∈ b[X ×Y, F ] if and only if φ(· , ·) = fj (·)gj (·), j  since J (Ω) = j J (fj ⊗ gj ) because J is linear. This characterizes the image ∗ ∗ of X ⊗∧ Y under the isometric isomorphism J : (X ⊗∧ Y)∗ → b[X ×Y, F ]:   J (X ∗ ⊗∧ Y ∗ ) = φ ∈ b[X ×Y, F ] : φ(·, ·) = j fj (·)gj (·) for every finite  sum with each fj in X ∗ and each gj in Y ∗ .  Y)∗ and for the  Y ∗ ⊆ (X ⊗ So (using the same notation twice) with Ω ∈ X ∗ ⊗ ∧ ∧  Y)∗ → b[X ×Y, F ] we get, by Theorem 7.9, isometric isomorphism J : (X ⊗ ∧    Y ∗ ) = φ ∈ b[X ×Y, F ] : φ(·, ·) = j fj (·)gj (·) for every countable J (X ∗ ⊗ ∧   sum with fj in X ∗ and gj in Y ∗ such that j fj gj  < ∞ . Corollary 7.14. Let X , Y, Z be arbitrary Banach spaces, let M and N be subspaces of X and Y, respectively, and consider the following assertions. (o) M and N are complemented in X and Y with M = R(E) and N = R(P ) for projections E ∈ B[X ] and P ∈ B[Y] such that E = P  = 1. (a) M ⊗∧ N is a linear manifold of X ⊗∧ Y. (b) Every bounded bilinear map φ : M×N → Z has a bounded bilinear exten = φ. sion φ : X ×Y → Z with φ (c) Every bounded linear transformation T : M ⊗∧ N → Z has a bounded linear extension T : X ⊗∧ Y → Z with T = T .  N → Z has a bounded lin(d) Every bounded linear transformation T : M ⊗ ∧    Y → Z with T  = T . ear extension T : X ⊗ ∧ The above assertions are related as follows: (o) =⇒ (a, b),

(a, b) ⇐⇒ (c) ⇐⇒ (d).

156

7. Norms on Tensor Products

Proof. Let X , Y, Z be arbitrary Banach spaces. (o) =⇒ (a,b). If (o) holds, then according to Theorems 6.8 and 7.8(d) (a) M ⊗∧ N is a linear manifold of X ⊗∧ Y,

and

(b) every φ ∈ b[M×N , Z] into an arbitrary Banach space Z has an  = φ. extension φ ∈ b[X ×Y, Z] with φ Now we proceed to show that (a,b), (c), and (d) are pairwise equivalent. Note that each extension in (c) or (d) only makes sense if (a) holds (since the linear transformation T|M ⊗∧ N is a restriction to linear manifold, and so is T|M ⊗  ∧N which acts on the completion of a linear manifold of X ⊗∧ Y). Let ∼ = stand for isometrically isomorphic. As we saw in the proof of Theorem 7.12, B[X ⊗ Y, Z] ∼ = b[X ×Y, Z]. ∧

The natural isometric isomorphism between them, J : B[X ⊗∧ Y, Z] → b[X ×Y, Z] such that ψ = J (S) for S ∈ B[X ⊗∧ Y, Z] and ψ ∈ b[X ×Y, Z], is given by ψ(x, y) = J (S)(x, y) = S(x⊗y) and so S(x⊗y) = J −1 (ψ)(x⊗y) = ψ(x, y) for every (x, y) ∈ X ×Y. By Theorem 6.8(a), M and N are complemented subspaces of X and Y if and only if every φ in b[M×N , Z] into an arbitrary Banach space Z has an extension φ in b[X ×Y, Z]. So in this case, with T in B[M ⊗∧ N , Z] and T in B[X ⊗∧ Y, Z] being the isometrically isomorphic images of φ in b[M×N , Z] and φ in b[X ×Y, Z], respectively, we get  M×N T ∼ = φ = φ|

and

φ ∼ = T.

Since the restriction T|M ⊗∧ N ∈ B[M ⊗∧ N , Z] of T ∈ B[X ⊗∧ Y, Z] to M ⊗∧ N only makes sense if (a) holds, then in this case with φ = J (T ) and φ = J (T),  M×N (u, v) = φ(u,  v) = J (T)(u, v) T (u ⊗ v) = J −1 (φ)(u ⊗ v) = φ(u, v) = φ| = T(u ⊗ v) = T|M ⊗ N (u ⊗ v) for every (u ⊗ v) ∈ M ⊗ N . ∧

∧

Hence T () = T|M ⊗∧ N () for every  =  M×N = φ φ|

 i

ui ⊗ vi ∈ M ⊗∧ N . Therefore

T|M ⊗∧ N = T. Thus if every φ ∈ b[M×N , Z] has an extension φ ∈ b[X ×Y, Z] and M ⊗∧ N is a linear manifold of X ⊗∧ Y, then every T ∈ B[M ⊗∧ N , Z] has an extension T ∈ B[X ⊗∧ Y, Z]. implies

A symmetric argument shows the converse. Suppose every T ∈ B[M ⊗∧ N , Z] has an extension T ∈ B[X ⊗∧ Y, Z]. Then by the above isometric isomorphisms

7.2 Projective Tensor Product

φ∼ = T = T|M ⊗∧ N

157

T ∼ = φ

and

whenever M ⊗∧ N is a linear manifold of X ⊗∧ Y. Hence φ(u, v) = J (T )(u, v) = T (u ⊗ v) = T|M ⊗∧ N (u ⊗ v)   v) = φ|  M×N (u, v) = T(u ⊗ v) = J −1 (φ)(u ⊗ v) = φ(u, for every (u, v) ∈ M×N . Therefore T|M ⊗∧ N = T

implies

 M×N = φ. φ|

Thus if every T ∈ B[M ⊗∧ N , Z] has an extension T ∈ B[X ⊗∧ Y, Z], then every φ ∈ b[M×N , Z] has an extension φ ∈ b[X ×Y, Z] and also M ⊗∧ N is a linear manifold of X ⊗∧ Y. Next assume one of the extensions in either (c) or (d) holds, which implies  N of M ⊗∧ N and X ⊗  Y of X ⊗∧ Y. By The(a). Take the completions M ⊗ ∧ ∧  Y. By Theorem 5.1(e), as Z is a  N is a subspace of X ⊗ orem 5.1(g), M ⊗ ∧ ∧  N , Z] and Banach space, B[M ⊗∧ N , Z] is isometrically isomorphic to B[M ⊗ ∧  B[X ⊗∧ Y, Z] is isometrically isomorphic to B[X ⊗∧ Y, Z]:  N , Z] ∼ B[M ⊗ = B[M ⊗∧ N , Z] ∧

and

 Y, Z] ∼ B[X ⊗ = B[X ⊗∧ Y, Z]. ∧

 N , Z] be the extension over completion Proceeding as before, let T in B[M ⊗ ∧ ∼  Y, Z] be the extension of T in B[M ⊗∧ N , Z] so that T = T , and let T in B[X ⊗ ∧ ∼    over completion of T in B[X ⊗∧ Y, Z] so that T = T . Extensions over completion are unique up to isometric isomorphism (cf. Theorem 5.1(c)). Hence T  Y, Z] extends T in B[M ⊗  N , Z] if and only if T in B[X ⊗∧ Y, Z] in B[X ⊗ ∧ ∧ extends T in B[M ⊗∧ N , Z]: T|M ⊗∧ N ∼ = T|M ⊗ 

∧

N

=T ∼ =T

and

T|M ⊗ 

∧

N

∼ = T. = T|M ⊗∧ N = T ∼

 N , Z] has an extension T ∈ B[X ⊗  Y, Z] if and Thus every T ∈ B[M ⊗ ∧ ∧ only if every T ∈ B[M ⊗∧ N , Z] has an extension T ∈ B[X ⊗∧ Y, Z].  = φ if and only if T = T . Finally, since φ ∼ = T and φ ∼ = T , then φ ∼ ∼   Similarly, since T = T and T = T , then T  = T if and only if T = T . Since the equivalence between the extensions in (b) under assumption (a) and (c), and the equivalence between the extensions in (c) and (d) had been verified before, then (a,b) ⇐⇒ (c) ⇐⇒ (d).   Corollary 7.14 is an application of Theorems 7.8 and 7.12 and, especially, an application of Theorem 6.8 on extension of bilinear maps. A very special particular case of Corollary 7.14 with a rather simplified statement is obtained by fixing Z = F . What is special in this particular case is this: it allows us to use the Hahn–Banach Theorem for the bounded linear functional in (c) whenever M ⊗∧ N is a linear manifold of X ⊗∧ Y.

158

7. Norms on Tensor Products

Corollary 7.15. If M and N are subspaces of Banach spaces X and Y, then the following assertions are pairwise equivalent. (a) M ⊗∧ N is a linear manifold of X ⊗∧ Y. (b) Every bounded bilinear form φ : M×N → F has a bounded bilinear exten = φ. sion φ : X ×Y → F with φ (c) Every bounded linear form f : M ⊗∧ N → F has a bounded linear extension f ∈ X ⊗∧ Y with f = f .  N → F has a bounded linear extension (d) Every bounded linear form f : M ⊗ ∧    Y → F with f  = f . f:X⊗ ∧ Proof. Let M and N be subspaces of Banach spaces X and Y. Equip tensor products with the projective norm, set Z = F , and recall from Remark 7.13(e):  Y)∗ ∼ (X ⊗ = (X ⊗∧ Y)∗ = B[X ⊗∧ Y, F ] ∼ = b[X ×Y, F ]. ∧  Take an arbitrary  = i ui ⊗ vi ∈ M ⊗∧ N . If assertion (b) holds, then every φ in b[M×N , F ] has an extension φ in b[X ×Y, F ] with the same norm. So the set of all bilinear forms in b[M×N , F ] with norm less than 1 is included in the set of all restrictions to M×N of all bilinear forms in b[X ×Y, F ] with norm less than 1. Then by Remark 7.13(f)     M ⊗∧ N = sup ψ(ui , vi )  i

ψ≤1, ψ∈b[M×N, F ]

≤ Since X

⊗∧ Y

sup ψ≤1, ψ∈b[X ×Y, F ]

    ψ(ui , vi ) = X  i

⊗∧ Y .

≤ M ⊗∧ N as we saw in Remark 7.5(b), then M ⊗∧ N = X

⊗∧ Y .

Equivalently, M ⊗∧ N is a linear manifold of X ⊗∧ Y by Remark 7.5(e). Thus (b) implies (a). The converse goes through (c). Actually, (a) implies (c) by the Hahn–Banach Theorem (and (c) only makes sense if (a) holds). Also, (c) implies (a,b) by setting Z = F in Corollary 7.14(c), and (a,b) in turn is equivalent to (b) (since (b) implies (a)), and so (c) implies (b). Finally (c) is equivalent to (d) by Corollary 7.14(c,d) with Z = F . Summing up, (b) =⇒ (a) =⇒ (c) =⇒ (b)

and

(c) ⇐⇒ (d).

 

By Remark 5.3, in a Hilbert-space setting each assertion in Corollaries 7.14 and 7.15 holds true (see also Proposition 5.R and Corollaries 5.6, 6.10). Remark 7.16. Immediate Consequences of Corollaries 7.14 and 7.15. (i) All norms in the proof of Corollary 7.14 are not only pairwise preserved but they coincide due to the isometric isomorphisms:

7.3 Injective Tensor Product

159

 = φ = T = T  = T = T . φ It is also worth noticing that extension over completion preserves norms so that T  = T  and T = T independently of Corollary 7.14. In particular, all norms in Corollary 7.15 coincide as well:  = φ = f = f  = f = f . φ (ii) By Theorem 5.13(a) and since Corollary 7.14 holds for every Banach space Z, any of the assertions (b), (c), (d) in Corollary 7.14 implies (o ) M and N are complemented. Moreover, according to Theorem 7.8(b) and Remark 7.5(e), if (o ) holds, then assertion (a) in both Corollaries 7.14 and 7.15 is equivalent to each of (a ) M ⊗∧ N is a subspace of X ⊗∧ Y,  N is a subspace of X ⊗  Y. (a ) M ⊗ ∧ ∧ (iii) The implication (o) =⇒ (b) in Corollary 7.14 has nothing to do with reasonable crossnorms. The equivalence (b) ⇐⇒ (c) in both corollaries holds for the projective norm only (by Theorem 7.12). The equivalence (c) ⇐⇒ (d) in both corollaries does not depend on the particular reasonable crossnorms. (iv) As we saw in Remark 7.5(b), M ⊗∧ N may not be a linear manifold of X ⊗∧ Y. Corollaries 7.14 and 7.15 gave equivalent conditions to ensure that (a) holds. However, as we also saw in Remark 7.5(e), if M ⊗∧ N is a linear  N is a subspace of X ⊗  Y. Regarding this manifold of X ⊗∧ Y, then M ⊗ ∧ ∧ particular issue, things get simpler for the injective tensor product: no extra  ∨N to be a subspace condition is needed for the injective tensor product M ⊗  of the injective tensor product X ⊗∨Y, as we will see in Theorem 7.19 below.

7.3 Injective Tensor Product Unlike the projective tensor product, there is no representation of elements  ∨Y: there is no injective counterpart of in the injective tensor product X ⊗ Theorem 7.9. There are, however, several interpretations of injective tensor  ∨Y of normed spaces X and Y by embedding them into Banach products X ⊗ spaces of bounded linear transformations or bounded bilinear forms following the patterns in Remark 4.9. Next we describe three such embeddings. Theorem 7.17. Let X and Y be normed spaces. (a1 ) If  ∈ X ⊗∨ Y, then ∨ = ψ b[X ∗ ×Y ∗, F ] where ψ ∈ b[X ∗ ×Y ∗, F ] is a  ∨Y defined by bounded bilinear form associated with  ∈ X ⊗  ψ (f, g) = f (xi ) g(yi ) for every (f, g) ∈ X ∗ ×Y ∗ , i

which does not depend on the (finite) representation

 i

xi ⊗ yi of .

160

7. Norms on Tensor Products

 ∨Y is isometrically embedded in the (a2 ) The injective tensor product X ⊗ Banach space b[X ∗×Y ∗, F ], and so is viewed as a subspace of b[X ∗×Y ∗, F ] : X ⊗∨ Y ⊆ b[X ∗ ×Y ∗, F ],

 ∨Y ⊆ b[X ∗ ×Y ∗, F ]. which implies X ⊗

(b1 ) If  ∈ X ∗ ⊗∨ Y, then  ∨ = Ψ B[X ,Y] , where Ψ ∈ B[X , Y] is a  ∨Y defined by bounded linear transformation associated with  ∈ X ∗ ⊗  fi (x) yi for every x ∈ X , Ψ x = i

which does not depend on the (finite) representation

 i

fi ⊗ yi of  .

 ∨Y is isometrically embedded in the (b2 ) The injective tensor product X ∗ ⊗ Banach space B[X , Y], and so it is viewed as a subspace of B[X , Y] :  ∨Y ⊆ B[X , Y] if Y is Banach. X ∗ ⊗∨ Y ⊆ B[X , Y], which implies X ∗ ⊗ (c1 ) If  ∈ X ⊗∨ Y ∗ , then  ∨ = Ψ B[Y,X ] , where Ψ ∈ B[Y, X ] is a  ∨Y ∗ defined by bounded linear transformation associated with  ∈ X ⊗  gi (y) xi for every y ∈ Y, Ψ y = i

which does not depend on the (finite) representation

 i

xi ⊗ gi of .

 ∨Y ∗ is isometrically embedded in the (c2 ) The injective tensor product X ⊗ Banach space B[Y, X ], and so it is viewed as a subspace of B[Y, X ] :  ∨Y ∗ ⊆ B[Y, X ] if X is Banach. X ⊗∨ Y ∗ ⊆ B[Y, X ], which implies X ⊗ Proof. (a) If X and Y are linear spaces, then X ⊗ Y ⊆ b[X  ×Y  , F ] by Remark 4.9(b). We will show that if X and Y are normed spaces and the tensor product X ⊗ Y is equipped with the injective norm  · ∨, then the above embedding is lifted to X ⊗∨ Y ⊆ b[X ∗ ×Y ∗, F ]. This is somewhat related to Corollary 4.10.  Let X and Y be normed spaces. Fix an arbitrary  = i xi ⊗ yi in X ⊗ Y. Consider the map ψ : X ∗ ⊗ Y ∗ → F associated with  defined by   ψ (f, g) = (f ⊗ g)() = f (xi ) ⊗ g(yi ) = f (xi ) g(yi ) i

i

for each (f, g) ∈ X ∗ ×Y ∗ . This does not depend on the representation of , by Theorem 3.17(a). As is readily verified, ψ is bilinear: ψ ∈ b[X ∗ ×Y ∗, F ]. Take the transformation K : X ⊗ Y → b[X ∗ ×Y ∗, F ] given by K() = ψ for every  in X ⊗ Y. This is clearly linear, and injective. (In fact, if K() = 0, then ψ = 0 so that ψ (f, g) = 0 for every (f, g) ∈ X ∗ ×Y ∗ and hence  = 0 by the Claim in the proof of Corollary 4.10.) Thus K is a linear-space embedding: K : X ⊗ Y → b[X ∗ ×Y ∗, F ],

which means

X ⊗ Y ⊆ b[X ∗ ×Y ∗, F ].

7.3 Injective Tensor Product

161

Now equip X ⊗ Y with the injective norm  · ∨. In this case ψ is bounded, that is, ψ ∈ b[X ∗ ×Y ∗ , F ] and, moreover (cf. Theorem 6.1),     sup sup f (xi ) g(yi ) = |ψ (f, g)| = ψ b[X ∗ ×Y ∗ ,F ] . ∨ =  f ≤1, g≤1

i

f ≤1,g≤1

As K()b[X ∗ ×Y ∗ ,F ] = ∨, the injective linear transformation K is an isometry and so K : X ⊗∨ Y → R(K) ⊆ b[X ∗ ×Y ∗, F ] is an isometric isomorphism onto its range R(K), which means K is an isometric embedding of X ⊗∨ Y into b[X ∗ ×Y ∗, F ], and so X ⊗∨ Y is interpreted as a linear manifold of b[X ∗ ×Y ∗, F ], X ⊗∨ Y ⊆ b[X ∗ ×Y ∗, F ]. The isometric isomorphism K ∈ B[X ⊗∨ Y, R(K)] extends over the completion  by Theo ∈ B[X ⊗  ∨Y of X ⊗∨ Y to an isometric isomorphism K  ∨Y, R(K)] X⊗ rem 5.1(d). Note: as b[X ∗ ×Y ∗, F ] is a Banach space, it is its own completion,  = R(K)−), and the completion of R(K) is its closure in b[X ∗ ×Y ∗, F ] (i.e., R(K)  is a subspace of b[X ∗ ×Y ∗, F ] (i.e., R(K)  ⊆ b[X ∗ ×Y ∗, F ]). Then and so R(K) ∗  K is an isometric embedding of X ⊗∨ Y into b[X ×Y ∗ , F ]), and therefore  ∨Y ⊆ b[X ∗ ×Y ∗, F ]. X⊗  ∨Y ∗ is included/isometrically In other words, the injective tensor product X ⊗ ∗ ∗ embedded in the Banach space b[X ×Y , F ], and in this sense it is regarded as a subspace (i.e., a closed linear manifold) of b[X ∗ ×Y ∗, F ]. (b) This follows a different but similar path from item (a). By Remark 4.9(i), X  ⊗ Y ⊆ L[X, Y] for linear spaces X and Y. Equip X ∗ ⊗ Y with the injective  ∨Y ⊆ B[X, Y] if Y norm and the inclusion is lifted to X ∗ ⊗∨ Y ⊆ B[X, Y] (so X ∗ ⊗ is Banach). Explanations have been given in (a) so we get straight to the point.  Equip X ∗ ⊗ Y with the injective norm  · ∨. Fix an arbitrary  = i fi ⊗ yi in X ∗ ⊗∨ Y, and consider the linear transformation Ψ : X → Y given by  fi (x) yi for every x ∈ X . Ψ x = i

(This also does not depend on the representation of  since for each x ∈ X , Ψ(·) (x) : X ∗ ⊗Y → Y is linear — same argument as in Theorem 3.17(a)’s proof.) Claim.

Ψ ∈ B[X , Y] and  ∨ = Ψ B[X ,Y] .

Proof of Claim. Let ϕ ∈ X ∗∗, g ∈ Y ∗ and x ∈ X be elements in X ∗∗, Y ∗ and X.    

     sup  ∨ = sup ϕ(fi ) g(yi ) = g(yi ) fi .  ϕ ϕ≤1,g≤1

i

ϕ≤1,g≤1

i

By the Claim in Theorem 7.3’s proof (a corollary of Goldstine’s Theorem), ϕ(f ) = limγ f (xγ )

with

xγ  ≤ ϕ

for every ϕ ∈ X ∗∗ , every f ∈ X ∗ , and some X -valued net {xγ }. Therefore

162

7. Norms on Tensor Products



 

ϕ g(yi ) fi = limγ g(yi )fi (xγ ) = limγ g(yi )fi (xγ ) i i i 

yi fi (xγ ) = limγ g(Ψ xγ ) = g(Ψ limγ xγ ). = limγ g i

Since ϕ = supf ≤1 |ϕ(f )| = supf ≤1 |f (limγ xγ )| =  limγ xγ ,  

   g(yi ) fi  = sup |g(Ψ x)|. sup ϕ i

ϕ≤1

x≤1

Moreover, since supg≤1 |g(y)| = y for every y ∈ Y and every g ∈ Y ∗ (by the Hahn–Banach Theorem — cf. Remark 5.2(d)),  

   sup sup g(yi ) fi  = |g(Ψ x)|= sup Ψ x = Ψ .   ϕ

ϕ≤1,g≤1

i

x≤1, g≤1

x≤1

The map  → Ψ of X ∗ ⊗∨ Y into B[X , Y] is clearly linear. It is an isometry (as seen above, thus injective). Then it is an isometric isomorphism onto its range, so an isometric embedding of X ∗ ⊗∨ Y into the normed space B[X , Y],  ∨Y of X ∗ ⊗∨ Y (Theorem 5.1(d)). which extends over the completion X ∗ ⊗  (c) This is a symmetric version of (b). Fix an arbitrary  = i xi ⊗ gi in X ⊗∨ Y ∗ and consider the linear transformation Ψ : Y → X given by  gi (y) xi for every y ∈ Y. Ψ y = i

Using the same argument as in the proof of item (b) we get Ψ ∈ B[Y, X ]

and

 ∨ = Ψ B[Y,X ] .

Thus the map  → Ψ is a linear isometry of X ⊗∨ Y ∗ into B[Y, X ], which  ∨Y ∗ is isometrically embedded in B[Y, X ] if X is Banach.   implies that X ⊗ Before proceeding, it is convenient to remark on a few additional points. Remark 7.18. Further Properties of the Injective Norm.  (a) Let X and Y be normed spaces. Take an arbitrary  = i xi ⊗ yi ∈ X ⊗ Y. If  = i xi ⊗ yi ∈ X ⊗∨ Y, then as we saw in proof of Theorem 7.17(a1 )     sup ∨ = sup f (xi )g(yi ) = |ψ (f, g)| = ψ b[X ∗ ×Y ∗, F ] ,  f ≤1, g≤1

i

f ≤1,g≤1

the injective norm ∨ of each  ∈ X ⊗∨ Y is the induced uniform norm ψ b[X ∗ ×Y ∗, F ] of the bounded bilinear map ψ ∈ b[X ∗×Y ∗, F ],  given by ψ (f, g) = i f (xi )g(yi ) for every (f, g) ∈ X ∗×Y ∗. The above expression implies another two expressions for  · ∨ on X ⊗ Y as in (b) and (c) below. (b) Since y = supg≤1 |g(y)| for every y ∈ Y and g ∈ Y ∗ (Remark 5.2(d)),

7.3 Injective Tensor Product

∨ =

sup

163

   

     f (xi )yi  = sup  f (xi )yi  = Ψ B[X ∗,Y] , g i

f ≤1, g≤1

f ≤1

i

Y

the injective norm ∨ of each  ∈ X ⊗∨ Y is the induced uniform norm Ψ B[X ∗,Y] of the bounded linear transformation Ψ ∈ B[X ∗, Y],  given by Ψ f = i f (xi )yi for every f ∈ X ∗. (c) Since x = supf ≤1 |f (x)| for every x ∈ X and f ∈ X ∗ (Remark 5.2(d)),    

     ∨ = sup g(yi )xi  = sup  g(yi )xi  = Ψ B[Y ∗,X ] , f i

f ≤1, g≤1

g≤1

i

X

the injective norm ∨ of each  ∈ X ⊗∨ Y is the induced uniform norm Ψ B[Y ∗,X ] of the bounded linear transformation Ψ ∈ B[Y ∗, X ],  which is given by Ψ g = i g(yi )xi for every g ∈ Y ∗. The next result justifies the terminology injective for the norm  · ∨. It shows that if M and N are linear manifolds of X and Y, respectively, then  ∨Y, thus embedded (injected) in X ⊗  ∨Y. In other  ∨ N is a subspace of X ⊗ M⊗ words, take a regular linear manifold M ⊗ N of the linear space X ⊗ Y. Equip each of them individually with the injective norm. Then M ⊗∨ N is a linear  ∨N manifold of X ⊗∨ Y (see Remark 7.5). Consequently, the completion M ⊗  ∨Y. is a subspace (i.e., a closed linear manifold) of the completion X ⊗ Theorem 7.19. If M and N are linear manifolds of normed spaces X and and Y, respectively, then M ⊗∨ N is a linear manifold of X ⊗∨ Y. Proof. Let M and N be linear manifolds of normed spaces X and Y, respectively. Consider the (algebraic) linear manifold M ⊗ N of X ⊗ Y. Equip these tensor products of normed spaces with the injective norm. We show that the norm identity in Remark 7.5(e) holds for the injective norm  · ∨. Therefore M ⊗∨ N is a linear manifold of the normed space X ⊗∨ Y (and consequently  ∨Y as we saw in Remark 7.5(d)). In fact, if X and  ∨N is a subspace of X ⊗ M⊗ Y are normed spaces and M and N are linear manifolds of X and Y (and so every u ∈ M and vN = vY for every v ∈ N ), then for uM = uX for  an arbitrary  = i ui ⊗ vi ∈ M ⊗∨ N we get, by Remark 7.18(b,c) (which was obtained by means of the Hahn–Banach Theorem as in Remark 5.2(d)),         f (ui )vi  = sup  f (ui )vi  = M ⊗∨ Y , M ⊗∨ N = sup  i i N Y f ≤1 f ≤1         M ⊗∨ Y = sup  g(vi )ui  = sup  g(vi )ui  = X ⊗∨ Y . g≤1

So M ⊗∨ N = X

i

⊗∨ Y

M

g≤1

i

X

. The claimed result follows by Remark 7.5(e).

Remark 7.20. Embeddings Related to Injective and Projective Norms. (a) According to Theorem 7.17,

 

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7. Norms on Tensor Products

X ⊗∨ Y ⊆ b[X ∗ ×Y ∗, F ],

X ∗ ⊗∨ Y ⊆ B[X , Y],

X ⊗∨ Y ∗ ⊆ B[Y, X ].

 ∨Y ∗ ⊆ B[X , Y ∗ ] ∩ B[Y, X ∗ ] (Theorem 5.1(g), By the last two inclusions, X ∗ ⊗ ∗ ∗ as B[X , Y ] and B[Y, X ] are Banach spaces) for the injective norm. By Re Y)∗ mark 6.4(a), B[X , Y ∗ ] ∼ = B[Y, X ∗ ] ∼ = b[X ×Y, F ]. Also, b[X ×Y, F ] ∼ = (X ⊗ ∧ for the projective norm (Theorem 7.12 for normed spaces X and Y). Thus  Y)∗ .  ∨Y ∗ ⊆ B[X , Y ∗ ] ∼ X∗⊗ = B[Y, X ∗ ] ∼ = b[X ×Y, F ] ∼ = (X ⊗ ∧ (Compare the embeddings involving the injective tensor product of the duals  ∨Y ∗ with those algebraic embeddings in Remark 4.9(b,f,g,h,i,).) SummaX∗⊗  ∨Y ∗ rizing the relation between the injective tensor product of the duals X ∗ ⊗ ∗  Y) of the projective tensor product X ⊗  Y: and the dual (X ⊗ ∧ ∧  Y)∗ ∼  ∨Y ∗ ⊆ (X ⊗ X∗⊗ = b[X ×Y, F ]. ∧ (b) We show that a similar relation holds if the injective tensor product of  ∨Y)∗ of the injective tensor  ∨Y ∗ is replaced by the dual (X ⊗ the duals X ∗ ⊗  ∨Y. In fact, for any normed space Z, let BZ = {z ∈ Z : zZ ≤ 1} product X ⊗ be the closed unit ball in Z (centered at the origin). Equip the tensor product X ⊗ Y with both injective and projective norms. That is, take the normed spaces X ⊗∨ Y = (X ⊗ Y,  · ∨) and X ⊗∧ Y = (X ⊗ Y,  · ∧). By Theorem 7.4  · ∨ ≤  · ∧

and so

X ⊗∧ Y ⊆ X ⊗∨ Y,

which is just a set inclusion: different norms on the same underlying linear space. Then ∧ ≤ 1 implies ∨ ≤ 1 for every  ∈ X ⊗∧ Y ⊆ X ⊗∨ Y. Therefore BX ⊗∧ Y ⊆ BX ⊗∨ Y , which in turn implies   (X ⊗∨ Y)∗ = f ∈ L[X ⊗ Y, F ] : sup∈BX ⊗ Y |f ()| < ∞ ∨   ⊆ f ∈ L[X ⊗ Y, F ] : sup∈BX ⊗ Y |f ()| < ∞ = (X ⊗∧ Y)∗ .

∧

 ∨Y)∗ and (X ⊗∧ Y)∗ ∼  Y)∗ by Theorem 5.1(f), Since (X ⊗∨ Y)∗ ∼ = (X ⊗ = (X ⊗ ∧  ∨Y)∗ ⊆ (X ⊗  Y)∗ ∼ (X ⊗ = b[X ×Y, F ]. ∧  ∨Y is embedded (algebraically) in the dual of Thus the dual of X ⊗  Y, which in turn is identified (isometrically) with b[X ×Y, F ]. X⊗ ∧  ∨Y)∗ is (algebraically) embedded in b[X ×Y, F ]. Then each element Hence (X ⊗  ∨Y)∗ is identified with a (unique) bilinear form φ in b[X ×Y, F ]. Λ in (X ⊗ The next result shows which class of bounded bilinear forms from b[X ×Y, F ]  ∨Y)∗ of the injective tensor product. represents all elements Λ in the dual (X ⊗ Theorem 7.21. A bounded bilinear form φ ∈ b[X ×Y, F ] on the Cartesian product of normed spaces X and Y is the identification of an element Λ from  ∨Y)∗ if and only if there is an F -valued measure μ on the Borel σ-algebra (X ⊗ of subsets of the compact space BX ∗×BY ∗ (where BX ∗ and BY ∗ are the closed unit balls in X ∗ and Y ∗ equipped with their weak∗ topology) such that

7.3 Injective Tensor Product

 φ(x, y) =

(f  ⊗ g  )(x, y) dμ(f  , g  )

BX ∗×BY ∗

for every

165

(x, y) ∈ X ×Y.

Before proving it we need four fundamental results as summarized below. (1) For any normed space Z let BZ ∗ = {f ∈ Z ∗ : f Z ∗ ≤ 1} be the closed unit ball in Z ∗, where f Z ∗ = supzZ ≤1 |f (z)|. The Banach–Alaoglu Theorem says that BZ ∗ is weak * compact (see, e.g., [70, Theorem 2.6.18]). Note that BZ ∗ is a Hausdorff space. Reason: every topological vector space (in particular, Z ∗ equipped with its weak* topology) is a Hausdorff space (see, e.g., [82, Theorem 1.12]), and a subspace of a Hausdorff space is Hausdorff (see, e.g., [19, Theorem 1.3]) — in this context, subspace means topological subspace (equipped with the induced topology). (2) Equip sets X and Y each with any topology. So their Cartesian product X×Y (equipped with the product topology) is compact if and only if X and Y are compact (Tychonoff Theorem — see, e.g., [19, Theorem XI.1.4(4)]). (3) Let Z be a topological space. With respect to the topology in Z and the usual (metric) topology in F (F is either R or C ) consider the linear space C[Z, F ] ⊆ F Z of all F -valued continuous functions defined on Z. If Z is compact, then every c ∈ C[Z, F ] is bounded in the sense that supz∈Z |c(z)| < ∞ (Weierstrass (little) Theorem — see, e.g., [19, Theorem XI.2.3]). (4) This is one of the general forms of the Riesz Representation Theorem. Let X be a compact Hausdorff space. If Γ : C[X, F ] → F is an F -valued bounded linear functional on C[X, F ], then there is a unique F -valued measure μ on the Borel σ-algebra XT of subsets of X such that  Γ (c) = c dμ for every c ∈ C[X, F ]. Moreover, Γ C[X,F ]∗ = |μ|(X) (see, e.g., [54, Theorems 12.6, 12.7]). Here |μ| is the total variation of μ, a positive finite (thus Borel) measure on XT — hence, unlike positive measures, F -valued measures (i.e., either real-valued measures, the so-called signed measures, or complex-valued measures) are naturally assumed to be Borel measures in the sense that their total variations are assumed to be finite and so Borel; and such a unique measure happens to be regular. Precisely, |μ| is the least positive finite measure dominating the F -valued measure μ (i.e., |μ| = min λ for all positive finite measures λ on |μ(E)| ≤ λ(E) for every E ∈ XT ) which also satisfies |μ|(X) = XT such that  sup|f |≤1 | f dμ| where the supremum is taken over all F -valued integrable functions f on X for which |f | ≤ 1 (see, e.g., [54, Chapter 10]). If X denotes the linear space of all F -valued measures on XT , then the function  · X : X → R given by μX = |μ|(X) is a norm on X, and so Γ C[X,F ]∗ = μX . Proof. Let C[BX ∗×BY ∗ , F ] denote the linear space of all F -valued continuous functions defined on the Cartesian product of the closed unit balls BX ∗×BY ∗ .

166

7. Norms on Tensor Products

∗ ∗ Take the transformation ψ ] → c ∈ C[BX ∗×BY ∗ , F ] such that  ∈ b[X ×Y , F  f g   c(f , g ) = ψ f X ∗ ) , gY ∗ with f = f fX ∗ ∈ BX ∗ and g  = ggY ∗ ∈ BY ∗ for (f, g) = 0 in X ∗ ×Y ∗. The transformation ψ → c is clearly linear, and injective (since c = 0 only if ψ = 0). Equip BX ∗ and BY ∗ with their weak* topologies. As they are compact, then so is BX ∗×BY ∗ . Hence the F -valued continuous functions on the compact set BX ∗×BY ∗ in C[BX ∗×BY ∗ , F ] are all bounded. Thus equip the linear space C[BX ∗×BY ∗ , F ] with the sup-norm  · ∞ so that

ψb[X ∗ ×Y ∗ ,F ] =

sup (f  ,g  )∈BX ∗×BY ∗

|c(f  , g  )| = c∞ .

Then the linear injective transformation ψ → c is an isometry, and so it is an isometric isomorphism onto its range, which means b[X ∗ ×Y ∗, F ] is isometrically embedded in C[BX ∗×BY ∗ , F ]. Let I be such an isometric embedding taking ψ ∈ b[X ∗ ×Y ∗, F ] to c = I(ψ) ∈ R(I) ⊆ C[BX ∗×BY ∗ , F ] ; that is, I : b[X ∗×Y ∗, F ] → C[BX ∗ ×BY ∗, F ], meaning b[X ∗×Y ∗, F ] ⊆ C[BX ∗ ×BY ∗, F ].  ∨Y is isometrically embedded in b[X ∗ ×Y ∗, F ] The injective tensor product X ⊗ according to Theorem 7.17(a2 ). Thus let K be that isometric embedding which  ∨Y to ψ = K() ∈ R(K) ⊆ b[X ∗ ×Y ∗, F ] so that takes  ∈ X ⊗  ∨Y → b[X ∗ ×Y ∗, F ], which means X ⊗  ∨Y ⊆ b[X ∗ ×Y ∗, F ]. K: X ⊗  ∨Y Therefore the composition IK is an isometric embedding taking  ∈ X ⊗ to c = I K() ∈ R(I K) ⊆ C[BX ∗×BY ∗ , F ] and so write  ∨Y → C[BX ∗×BY ∗ , F ], meaning X ⊗  ∨Y ⊆ C[BX ∗×BY ∗ , F ]. I K: X ⊗  ∨Y)∗ = B[X ⊗  ∨Y, F ], and consider the above incluTake an arbitrary Λ ∈ (X ⊗ sion/embedding. By the Hahn–Banach Theorem (cf. Remark 5.2(d)), we can extend Λ to Λ ∈ B[C[BX ∗×BY ∗ , F ], F ] = C[BX ∗×BY ∗ , F ]∗ with norm preservation. By the Riesz Representation Theorem, there is an F -valued measure μ on the Borel σ-algebra of subsets of the compact set BX ∗×BY ∗ such that    Λ(c) = c dμ = c (f  , g  ) dμ(f  , g  ) for every c ∈ C[BX ∗×BY ∗ , F ]. BX ∗×BY ∗

 ∨Y ⊆ C[BX ∗×BY ∗ , F ], In particular, for  ∈ X ⊗   ) =  dμ for every Λ() = Λ(

 ∨Y. ∈X⊗

 ∨Y, More particularly, for an arbitrary single tensor x ⊗ y ⊆ X ⊗∨ Y ⊆ X ⊗  Λ(x ⊗ y) = x ⊗ y dμ for every (x, y) ∈ X ×Y. Consider again the isometric embedding K of Theorem 7.17(a1 ,a2 ). Thus each single tensor x ⊗ y in X ⊗∨ Y is identified with a bounded bilinear form ψx⊗y in b[X ∗ ×Y ∗ , F ] given by ψx⊗y (f, g)=f (x) g(y) for every (f, g) in X ∗ ×Y ∗ . So

7.4 Additional Propositions



167





ψx⊗y (f  , g  ) dμ(f  , g  ) x ⊗ y dμ = ψx⊗y dμ = BX ∗×BY ∗  = f  (x)g  (y) dμ(f  , g  ) = φ(x, y) for every (x, y) ∈ X ×Y, BX ∗×BY ∗

defining a bounded bilinear form φ : X ×Y → F (i.e., a form φ ∈ b[X ×Y, F ]).  ∨Y)∗ is algebraically embedded in b[X ×Y, F ] (cf. Remark 7.20(b)), As (X ⊗  ∨Y)∗ is regarded as an algebraic linear manifold of b[X ×Y, F ], and (X ⊗ Λ(x ⊗ y) = φ(x, y)

for every

x ∈ X and y ∈ Y

 ∨Y)∗ → φ ∈ b[X ×Y, F ] for establishes the claimed identification: Λ ∈ (X ⊗   φ(x, y) = (f  ⊗ g  )(x, y) dμ(f  , g  ) for every (x, y) ∈ X ×Y.  BX ∗×BY ∗

Note that, since norms in the above proof are preserved by the Riesz representation and by the Hahn–Banach extension, and since φ has been identified with the restriction of Λ to single tensors, we get  C[B ∗×B ∗ ,F ]∗ = Λ  ∗ and μX = Λ (X ⊗∨Y) X Y

φb[X ×Y,F ] = sup

x⊗y=0

|Λ(x⊗y)| x⊗y∨ .

7.4 Additional Propositions Throughout this section X , Y, Z are nonzero Banach spaces over a field F . Equip the tensor product space X ⊗ Y with the injective and projective norms  · ∨ and  · ∧. Set X ⊗∨ Y = (X ⊗ Y,  · ∨) and X ⊗∧ Y = (X ⊗ Y,  · ∧).  Y are the injective and the projective  ∨Y and X ⊗ Their completions X ⊗ ∧ tensor products. As before, ∼ = means isometrically isomorphic. By a scalar sequence (function) we mean a scalar-valued sequence (function). Proposition 7.A. X ⊗∧ Y is complete ⇐⇒ X or Y is finite-dimensional . The next two results exhibit a Banach-space projective version of the algebraic results in Corollary 3.11(a) and Proposition 3.D(c). ∼ Y⊗  X.  Y= Proposition 7.B. X ⊗ ∧

∧

 Y) ⊗  Z∼  (Y ⊗  Z). Proposition 7.C. (X ⊗ =X⊗ ∧ ∧ ∧ ∧ By a closed unit ball we mean one centered at the origin of a normed space. Also, in light of Theorem 3.13(b), if X and Y are linear spaces and if M ⊆ X and N ⊆ Y are subsets (but not linear manifolds) of X and Y, respectively, then by M ⊗ N we mean the range of θ|M ×N , where θ : X ×Y → X ⊗ Y is the natural bilinear map associated with the tensor product space X ⊗ Y. Proposition 7.D. Let X and Y be Banach spaces and let BX and BY  Y is denote the closed unit balls in X and Y. The closed unit ball in X ⊗ ∧ (isometrically isomorphic to) the closed convex hull of BX ⊗ BY .

168

7. Norms on Tensor Products

m Set F m (Z) = Z m = i=1 Z = {(z1 , . . . , zm ) : zi ∈ Z, i = 1, . . . , m} for any linear space Z (cf. Sections 1.1 and 4.4). Proposition 3.F says that F m ⊗ Z = F m (Z) for every linear space Z. The next result goes along this line extending from F m to 1+ . If Z is a Banach space, then let 1+ (Z) denote the Banach space of all absolutely summable  Z-valued sequences {zk } where the norm on 1+ (Z) is given by {zk }1 = k zk . For Z = F we get the standard Banach space 1+ = 1+ (F ) of all absolutely summable scalar sequences.  Z∼ Proposition 7.E. 1+ ⊗ = 1+ (Z). ∧  1+ ∼ Proposition 7.F. 1+ ⊗ = 1+ (1+ ) ∼ = 1+ . ∧ If M and N are subspaces of X and Y, then M ⊗∧ N may not be a linear manifold of X ⊗∧ Y under the projective norm (cf. Remark 7.5). Here are two examples where the projective tensor product of subspaces yields a subspace. The next result is a relevant application of Proposition 6.J from Section 6.4.  Y ∗∗.  Y is a subspace of X ∗∗ ⊗ Proposition 7.G. X ⊗ ∧ ∧  M is a subspace of 1+ ⊗  Z if M is a subspace of Z. Proposition 7.H. 1+ ⊗ ∧ ∧ Proposition 7.E is a particular case of Proposition 7.I below for the counting measure. Let L1 (Z) = L1 (μ, Z) = L1 (Ω, A, μ; Z) stand for the Banach space of all (equivalence class of) Z-valued integrable functions with respect to a measure space (Ω, A, μ) (where μ is a measure on a σ-algebra A of subsets of a nonempty set Ω), and by a Z-valued integrable function F:Ω →Z  we mean it is integrable in the Bochner sense with F 1 = F  dμ (see, e.g., [54, Proposition 10.H]). Again, for Z = F we get the standard Banach space L1 = L1 (μ) of all integrable scalar functions.  Z∼ Proposition 7.I. L1 ⊗ = L1 (Z). ∧  Y → Z, The graph GL of a linear transformation L : X ⊗ ∧    Y)×Z : z = L() , GL = (, z) ∈ (X ⊗ ∧  Y)×Z. Equip the Cartesian product is a subset of the Cartesian product (X ⊗ ∧  Y and in Z (as in with a product metric d generated by the norms in X ⊗ ∧ Propositions 5.D and 6.F in Sections 5.4 and 6.4). The Closed Graph Theorem still holds for linear transformations on projective tensor products as follows.  Y → Z is linProposition 7.J. If X , Y, Z are Banach spaces and T : X ⊗ ∧ ear, then T is continuous if and only if its graph GT is closed . Up to this point all propositions in this section dealt with the projective norm. The next four propositions deal with the injective norm. For a Banach space Z let c0 (Z) be the Banach space of all Z-valued sequences converging to zero under the sup-norm. For Z = F we get the standard Banach space c0 = c0 (F ) of all scalar sequences converging to zero.

7.4 Additional Propositions

169

 ∨Z ∼ Proposition 7.K. c0 ⊗ = c0 (Z).  ∨c0 ∼ Proposition 7.L. c0 ⊗ = c0 . = c0 (c0 ) ∼ Let K be a compact set in a Hausdorff space, let C(K) denote the Banach space of all F -valued continuous functionals on K equipped with the sup-norm  · ∞ , and let C(K, Z) be the Banach space of all Z-valued continuous mappings F : K → Z on K equipped with the sup-norm (F ∞ = supx∈K F (x)Z ).  ∨Z ∼ Proposition 7.M. C(K) ⊗ = C(K, Z).  ∨Y ∼  ∨X . Proposition 7.N. X ⊗ = Y⊗ Back to the projective norm, consider the dual of the projective tensor product. Theorem 7.12 ensures the existence of an isometric isomorphism  Y)∗ → b[X ×Y, F ]. J : (X ⊗ ∧  Y)∗ ∼  Y ∗ ⊆ (X ⊗ Since X ∗ ⊗ = b[X ×Y, F ] (cf. Remark 7.13(g)), take the re∧ ∧  Y ∗ . As we saw in Remark 7.13(g), the striction of J to the linear space X ∗ ⊗ ∧ above inclusion ⊆ means linear-space embedding but not normed-space em Y ∗ is not regarded as a linear manifold of the normed bedding, and so X ∗ ⊗ ∧ ∗  Y) (they are equipped with different norms). Thus set space (X ⊗ ∧ J = J |X ∗ ⊗ 

∧

Y∗

 Y ∗ → b[X ×Y, F ]. : X∗⊗ ∧

 Y ∗ → b[X ×Y, F ] has the following properties. Proposition 7.O. J : X ∗ ⊗ ∧ (a) J is linear . (b) J is a contraction with J = 1. (c) φ ∈ b[X ×Y, F ] lies in the range R(J) of J if and only if there are X ∗valued and Y ∗-valued sequences {fk } and {gk } for which the real sequence {fk gk } is summable and  φ(x, y) = fk (x)gk (y) for every (x, y) ∈ X ×Y. k

(d) For each φ ∈ R(J), φN =

inf

{fk },{gk }, φ=Σk fk (·) gk (·)

 k

fk gk 

defines a norm on the linear space R(J) for which φ ≤ φN . The range R(J) of J is referred to as the linear space of nuclear bilinear forms. Thus set bN [X ×Y, F ] = R(J) ⊆ b[X ×Y, F ]. A bilinear form φ is nuclear (i.e., φ ∈ bN [X ×Y, F ]) if and only if ∗ ∗ there  sequences {fk } and {gk } such that  are X -valued and Y -valued f g  < ∞ and φ(x, y) = k k k k fk (x)gk (y) for every (x, y) ∈ X ×Y.  The expression φ(x, y) = k fk (x)gk (y) is a nuclear representation of φ, and

170

7. Norms on Tensor Products

φN = inf

 k

fk gk  < ∞

φ ∈ bN [X ×Y, F ]

for every

(where the infimum is taken over all nuclear representations of φ) defines a norm on the linear space bN [X ×Y, F ]. This norm dominates the usual norm on b[X ×Y, F ] (i.e., φ ≤ φN ), and (bN [X ×Y, F ],  · N ) is a Banach space. The linear contraction J of norm one in Proposition 7.O is not necessarily injective (i.e., N (J) is not necessarily equal to {0}), so J is not necessar Y ∗ onto R(J) = ily isometric, nor is it necessarily an isomorphism of X ∗ ⊗ ∧  Y∗ bN [X ×Y, F ]. In fact, R(J) is identified with the quotient space of X ∗ ⊗ ∧ ∗ ∗  Y )/N (J) (see, e.g., [83, p. 40]). modulo N (J), that is, bN [X ×Y, F ] ∼ = (X ⊗ ∧ From Theorem 7.17 we know that there are several interpretations of injective tensor products by embedding them into Banach spaces of bounded linear transformations following the patterns in Remark 4.9. In particular, we saw  ∨Y ⊆ B[X , Y]. This isometric embedding does in Theorem 7.17(b2 ) that X ∗ ⊗ not hold in general for the projective norm. The result below shows how far one can get along this line for the projective norm in the general case.  Y → B[X , Y] such that Proposition 7.P. There is a transformation K : X ∗ ⊗ ∧ (a) K is a linear contraction with K = 1, (b) whose range R(K) is characterized as follows : T ∈ B[X , Y] lies in R(K) if and only if there are X ∗-valued and Y-valued sequences {fk } and {yk } for which the real sequence {fk yk } is summable and  Tx = fk (x)yk for every x ∈ X , k

(c) and for each T ∈ R(K), T N =

inf

{fk },{yk }, T =Σk fk (·) yk

 k

fk yk 

defines a norm on the linear space R(K) for which T  ≤ T N . A transformation T ∈ B[X , Y] is nuclear if it lies in the range R(K) of the  Y, B[X , Y]] defined in Proposition 7.P, and the range contraction K ∈ B[X ∗ ⊗ ∧ R(K) itself is called the linear space of nuclear transformations. Thus set BN [X , Y] = R(K) ⊆ B[X , Y].

In particular, BN [X ] = BN [X , X ] ⊆ B[X ].

A linear transformation T is nuclear (i.e., T ∈ BN [X , Y]) if and only ∗ if there  sequences {fk } and {yk } such  exist X -valued and Y-valued that k fk yk  < ∞ and T x = k fk (x)yk for every x ∈ X .

T N



fk (x)yk is a nuclear representation of T , and  = inf fk yk  for every T ∈ BN [X , Y]

The expression T x =

k

k

7.4 Additional Propositions

171

(where the infimum is taken over all nuclear representations of T ) defines a norm on the linear space BN [X , Y], the nuclear norm, such that T  ≤ T N . As before, (BN [X , Y],  · N ) is a Banach space, the linear contraction K of norm one in Proposition 7.P is not necessarily injective — thus K may not  Y be an isometry — and R(K) is identified with the quotient space of X ∗ ⊗ ∧ ∗ ∼ modulo N (K), that is, BN [X , Y] = (X ⊗∧ Y)/N (K) (see, e.g., [83, p. 41]). It is worth noting the following special case. Suppose Y is such that Y ∼ = V∗ for some Banach space V (for instance, if Y is reflexive). By Theorems 7.12 and  V)∗ ∼ 6.3 we get (X ⊗ = b[X ×V, F ] ∼ = B[X , V ∗ ]. Moreover, by Definition 7.1 the ∧ ∗ ∗ linear space X ⊗ V is a linear manifold of the linear space (X ⊗∧V)∗, which is  V)∗ by Theorem 5.1(f). If equipped with the isometrically isomorphic to (X ⊗ ∧ projective norm, the linear space X ∗ ⊗∧V ∗ remains algebraically embedded in  V)∗ (with different norms, see Remark 7.13(g)). So the Banach space (X ⊗ ∧  V)∗ ∼  Y∼  V ∗ ⊆ (X ⊗ X∗⊗ = X∗⊗ = B[X , V ∗ ] ∼ = B[X , Y ], ∧ ∧ ∧ where ⊆ means algebraic embedding and ∼ = isometric isomorphism. Therefore Y∼ = V ∗ for some V

=⇒

 Y ⊆ B[X , Y]. X∗⊗ ∧

 V)∗ → B[X , Y], the linear Since there is an isometric isomorphism K : (X ⊗ ∧ ∗ ∗ ∗ ∼ contraction K : X ⊗∧ Y = X ⊗∧V → B[X , Y] of Proposition 7.P can in this  Y: case be taken as the restriction of the isometric isomorphism K to X ∗ ⊗ ∧ K = K|X ∗ ⊗ 

∧

Y

 Y → B[X , Y]. : X∗⊗ ∧

Thus if Y ∼ = V ∗ (i.e., if the Banach space Y is the dual of a Banach space V), then Proposition 7.P has a proof following the argument of Proposition 7.O. Proposition 7.Q. Let X , Y, V, W be Banach spaces, Suppose T ∈ BN [X , Y], Then for every R ∈ B[V, X ] and L ∈ B[Y, W] it follows that LT R ∈ BN [V, W] and L T RN ≤ LT N R. This shows that BN [X ] is an ideal of B[X ]. Also, BN [X , Y] ⊆ B∞ [X , Y] (nuclear transformations are compact, as a consequence of Proposition 5.H). From now on let X be a Hilbert space, denote its inner product by · ; ·, and take the Banach algebra B[X ] of all operators on X . Consider the Fourier Series and the Riesz Representation Theorems (Propositions 5.T and 5.U). A functional f lies in X ∗ if and only if there exists a unique z in X (which does not depend on x) such that f (x) = x ; z for every x ∈ X and z = f . This unique z is the Riesz representation of f . Thus a nuclear operator acting on a Hilbert space is defined as follows. On a Hilbert space X an operator T ∈ B[X ] is nuclear (i.e., T ∈ BN [X ]) if  sequences {zk } and {yk } such that and only if there are X -valued z y  < ∞ and T x = k k k k x ; zk yk for every x ∈ X . For each T ∈ B[X ] let T ∗ ∈ B[X ] stand for its Hilbert-space adjoint.

172

7. Norms on Tensor Products

Proposition 7.R. Let T ∈ B[X ] be an operator on a Hilbert space X . (a) If there exist X -valued sequences {zk } and {yk } such that  x ; zk yk for every x ∈ X , Tx = k

then

T ∗x =

 k

x ; yk zk

x ∈ X.

for every

(b) T is nuclear if and only if its adjoint T ∗ is nuclear and T ∗ N = T N . For each operator T ∈ B[X ] on a Hilbert space X, set |T | = (T ∗ T )1/2 ∈ B[X ]. Take an arbitrary orthonormal basis  {eγ } for X . Consider the family of nonnegative numbers {|T |eγ ; eγ }. If γ |T |eγ ; eγ  < ∞, then its value in R does not depend on the choice  of the orthonormal basis {eγ }. An operator T ∈ B[X ] is called trace-class if γ |T |eγ ; eγ  < ∞ for any orthonormal basis {eγ }. This  implies that γ |T eγ ; eγ | < ∞ for every orthonormal basis {eγ }. In this case {T eγ ; eγ } is a summable family of scalars, and its sum is denoted by tr(T ) =  γ T eγ ; eγ  in F and referred to as the trace of T (which again does not depend on the choice of the orthonormal basis). Let B1 [X ] denote the collection of all trace-class operators from B[X ], which is a linear manifold of the linear space B[X ]. Actually, B1 [X ] is an ideal of the algebra B[X ], and every operator in B1 [X ] is compact (i.e., B1 [X ] ⊆ B∞ [X ]). For each T ∈ B1 [X ], set   T 1 = tr(|T |) = |T |eγ ; eγ  = |T |1/2 eγ 2 . γ

k

The function  · 1 : B1 [X ] → R is a norm on the linear space B1 [X ], the tracenorm, and T  ≤ T 1 . (B1 [X ],  · 1 ) is a Banach space. Also, T ∈ B1 [X ] if and only if |T | ∈ B1 [X ] and |T |1 = T 1 . Moreover, T ∈ B1 [X ] if and only if T ∗ ∈ B1 [X ] and T ∗ 1 = T 1 . (See, e.g., [52, Problems 5.64 to 5.69].) Proposition 7.S. Suppose X is a Hilbert space. (a) B1 [X ] = BN [X ] and  · 1 =  · N . Take an arbitrary operator T ∈ B[X ].  (b) T ∈ B1 [X ] ⇐⇒ γ T eγ  < ∞ for some orthonormal basis {eγ } for X .  (c) T ∈ B1 [X ] =⇒ γ |T eγ ; eγ | < ∞ for every orthonormal basis {eγ } for X . (d) The converse of (c) fails.  ∗ (As γ T eγ  < ∞ in (b) by  T∗∈ B1 [X ] ⇐⇒ T∈ B1 [X ], we may replace  ∗ T e  < ∞ and |T e ; e | < ∞ in (c) by γ γ γ γ γ γ |T eγ ; eγ | < ∞.) Notes: The first two paragraphs of the Notes in Sections 3.4 and 4.4 summarized a little of the tensor product saga since Schatten’s [85] monograph in 1950, which resulted from his thesis as of 1942. Thus such a tensor product theory is indeed a very young child. In 1953 the theory was further developed

7.4 Additional Propositions

173

by Grothendieck [28, 29] in his celebrated R´esum´e. This chapter gave a brief introduction to the notion of reasonable crossnorms, highlighting the elementary properties of their extremes, namely, the projective and injective norms. A question which goes further, much further, than the commutation properties in Propositions 7.B and 7.N asks whether commutativity survives with mixed norms. It is clear by Corollary 3.10 that if X and Y are finite-dimensional normed spaces (where all norms are equivalent and normed spaces are complete), then X ⊗∨ Y ∼ = Y ⊗∧X . Grothendieck left as an open question whether  X holds for some pair of infinite-dimensional Banach spaces X  ∨Y ∼ X⊗ =Y⊗ ∧ and Y. A solution was given by Pisier [73, 74], who exhibited in 1981 a separable infinite-dimensional Banach space P, now called Pisier space, such that  P.  P∼ P⊗ = P⊗ ∨

∧

This is a seminal result in itself, which shows in addition that all reasonable crossnorms on P ⊗ P are equivalent. Proposition 7.A explains why completions are required, and Propositions 7.B and 7.C show that some elementary algebraic properties survive when the tensor product spaces are equipped with the projective norm (see, e.g., [83, Exercises 2.2–2.5] — Proposition 7.B also follows from Remark 7.7(d)). For Proposition 7.D see, e.g., [83, Proposition 2.3]. As we saw in Section 4.4, tensor products of finite-dimensional linear spaces are identified with Kronecker products (the identification there is attained by algebraic isomorphism ∼ =), and so F m ⊗ F n ∼ = = F m (F n ). This is generalized in Proposition 3.F to F m ⊗ Z ∼ F m (Z) for any linear space Z, and it survives isometrically the extension from F m to the infinite-dimensional space 1+ = (1+ ,  · 1 ) when tensor products  Z ∼ are normed with the projective norm  · ∧, viz., 1+ ⊗ = 1+ (Z) for every ∧ Banach space Z as in Proposition 7.E (see, e.g., [83, Example 2.6]). Proposition 7.F is a consequence of Proposition 7.E (see also [83, Exercise 2.6]). Proof of Proposition 7.G. Apply Proposition 6.J (Section 6.4) and Corollary 7.15 to verify that X ⊗∧ Y is a linear manifold of X ∗∗ ⊗∧ Y ∗∗ , and therefore  Y is a subspace of X ∗∗ ⊗  Y ∗∗ by Theorem 5.1(g).   X⊗ ∧ ∧  M∼ Proof of Proposition 7.H. As M and Z are Banach spaces, 1+⊗ = 1+ (M) ∧ 1  1 and +⊗∧Z ∼ = + (Z) by Proposition 7.E. If M is a subspace of Z, then 1+ (M)  M is isometrically embedded in 1+⊗  Z.   is a subspace of 1+ (Z), and so 1+⊗ ∧ ∧ For Proposition 7.I see, e.g., [17, Theorem 1.1.10]. Proposition 7.J follows from the Closed Graph Theorem for bounded bilinear maps in Proposition 6.F by using the universal mapping property of Theorem 7.12 (see, e.g., [15, Exercise 3.10]). Proposition 7.K goes along the line of Proposition 7.E, now with respect to the injective norm  · ∨ (see, e.g., [17, Theorem 1.1.11]). Proposition 7.L follows partially from Proposition 7.K (see, e.g., [83, Exercise 3.1]). Proposition 7.M can be viewed as the injective counterpart of Proposition 7.I (see, e.g., [17, Theorem 1.1.10]). Remark 7.7(d) implies Proposition 7.N.

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7. Norms on Tensor Products

Proof of Proposition 7.O. (a) Since a restriction of a linear transformation to a linear manifold of its domain is again linear, J = J |X ∗ ⊗  Y ∗ is linear. ∧

 Y)∗ . Consider the isometric iso Y ∗ ⊆ (X ⊗ (b) Take an arbitrary Ω ∈ X ∗ ⊗ ∧ ∧ ∗  Y) → b[X ×Y, F ]. Since Ω∗∧ ≤ Ω∧ (Remark 7.13(g)), morphism J : (X ⊗ ∧ J = J |X ∗ ⊗ 

∧

Y∗ 

=

sup J (Ω) =

Ω∧≤1

sup Ω∗∧ ≤

Ω∧≤1

sup Ω∧ = 1.

Ω∧≤1

 Y)∗  Y ∗ ⊆ (X ⊗ Take (f, g) ∈ X ∗×Y ∗ with f  = g = 1 so that f ⊗ g ∈ X ∗ ⊗ ∧ ∧ is such that f ⊗ g∧ = 1. As we saw in Remark 7.13(h), φ = J(f ⊗ g) = J (f ⊗ g) in b[X ×Y, F ] is such that φ(x, y) = f (x)g(y) for every (x, y) ∈ X ×Y.  Y∗ So J(f ⊗ g) = φ = f g = 1. Hence there exists Ω = f ⊗ g ∈ X ∗ ⊗ ∧ with Ω∧ = 1 such that J(Ω) = 1. Then J ≥ 1. Therefore J = 1.  Y ∗ ). (c) This was proved in Remark 7.13(h) since R(J) = J (X ∗ ⊗ ∧ (d) It is readily verified that  · N is a norm on triangle  R(J). We show  the     ∈ R(J) so that φ + φ = f (·)g (·)+ f (·)g inequality. Take φ, φ k k k k (·) = k k          f (·)g (·). Thus φ + φ  = inf ( f g  + f g N k k k k k ) = k k  k  fk ,gk ,fk ,gk inf fk ,gk k fk gk  ≤ inf fk ,gk k fk gk  + inf fk ,gk k fk gk  =  Y ∗ ), then φN + φ N . As for the norm inequality, if φ ∈ R(J) = J (X ∗ ⊗ ∧ ∗ ∗ ∗  Y) such that φ = J (Ω). So (Remark 7.13(g)), there is Ω ∈ X ⊗∧ Y ⊆ (X ⊗ ∧  φ = J (Ω) = Ω∗∧ ≤ Ω∧ = inf fk gk  = φN .   k Proof of Proposition 7.P. Let X ∗ be the dual of X . By Theorem 7.9, take an    Y so that ∧ = inf k fk yk . Assoarbitrary  = k fk ⊗ yk in X ∗ ⊗ ∧ ciated with , consider the transformation Ψ : X → Y given by  Ψ x = fk (x) yk for every x ∈ X , k

 which is clearly linear and does not depend on the representation k fk ⊗ yk of  (same argument as in Theorem 3.17(a)’s proof). Also, Ψ is bounded. In  fact, Ψ x ≤ k fk yk x, so that Ψ x ≤ ∧x, for every x ∈ X .  Y. This defines a transformation Thus Ψ lies in B[X , Y] for each  ∈ X ∗ ⊗ ∧  Y → B[X , Y] K: X∗⊗ ∧

 Y. K() = Ψ for every  ∈ X ∗ ⊗ ∧     Indeed,  +  = k fk ⊗ yk + k fk ⊗ yk , (a1 ) It is clear thatthis K is linear. so that Ψ+ x = k fk (x) ⊗ yk + k fk (x) ⊗ yk = Ψ x + Ψ x for each x, and hence K( +  ) = Ψ+ = Ψ +Ψ = K()+K( ). Similarly, K(α) = α. such that

(a2 ) K is a contraction. In fact, as Ψ x ≤ ∧x for every x ∈ X, K() =  Y. So Ψ  = supx=1 Ψ x ≤ supx=1 ∧x = ∧ for every  ∈ X ∗ ⊗ ∧   ∗  Y, B[X , Y] K∈B X ⊗ ∧  Y with f  = with K = sup∧=1 K() ≤ 1. Reversely, for f ⊗ y in X ∗ ⊗ ∧ y = 1 we get Ψf ⊗y x = |f (x)|y = |f (x)|. Hence K(f ⊗ y) = Ψf ⊗y  =

7.4 Additional Propositions

175

supx=1 |f (x)| = f  = 1 = f y = f ⊗ y∧. Then there exists  = f ⊗ y in X ∗ ⊗∧ Y with ∧ = 1 for which K() = ∧. So K ≥ 1. Thus K = 1. (b) By definition K(f ⊗ y) = T ∈ B[X , Y] if and  only if T x = Ψf ⊗y (x) = f (x)y  Y, for every x ∈ X . Then for an arbitrary  = k fk ⊗ yk ∈ X ∗ ⊗ ∧ K() = T ∈ B[X , Y] if and only if T x = Ψ (x) = fk (x)yk k  for every x ∈ X , since K() = k K(fk ⊗ yk ) because K is linear and bounded. This characterizes the range R(K) of K. (c) It is easy to see that  · N is a norm on R(K). inWe show the triangle     ∈ R(K) so that T + T = f (·)y + f (·)y equality only. Take T, T k k k = k k  k         f (·)y . Thus T + T  = inf ( f y  + f y ) = N k k k k k k k fk ,yk ,fk ,yk  k inf fk ,yk k fk yk  ≤ inf fk ,yk k fk yk  + inf fk ,yk k fk yk  = by (b) there T N + T  N . To verify the norm inequality, if T ∈ R(K), then  {f y } summable such that T (x) = are {fk } and {yk } with k k k fk (x)yk ,  and hence T x =  k fk (x)yk , for every x ∈ X . Therefore   fk yk x = fk yk  T  = sup T x ≤ sup x=1

so that T  ≤ inf

k

x=1

 k

k

fk yk  = T N .

 

Proof of Proposition 7.Q. This is immediate by definition,  and it shows that is a two-sided ideal of the algebra B[X ]. Since k fk y BN [X ]  k  < ∞ and ∗ f (x)y , set g = f R ∈ V and w = Ly ∈ W. So T x = k k k k k k k   k gk yk  ≤  f y R < ∞ and T Rx = g (x)y , as well as k k k k k k k fk wk  ≤     k fk yk L < ∞ and LT x = k fk (x)wk . Proof of Proposition 7.R. Take an operator T ∈ B[X ] on a Hilbert space X . (a) Suppose there are X -valued sequences {zk } and {yk } such that  x ; zk yk for every x ∈ X . Tx = k

      Thus T x ; y = x ; y ; y = x ; z y ; y = x ; z y ; y = z k k k k k k k k k    x ; k y ; yk zk for every x, y ∈ X , and hence (by uniqueness of the adjoint)  T ∗y = y ; yk zk for every y ∈ X . k

(b) An operator T lies in  BN [X ] if and only if there exist  X -valued sequences {zk } and {yk } such that k zk yk  < ∞ and T x = k x ; zk yk for every x ∈ X . Therefore if T lies in BN [X ], then T ∗ lies in BN [X ] by item (a). with the infimum taken over The converse holds since T ∗∗ = T . Moreover,  all sequences {yk } and {zk } such that k zk yk  < ∞,  yk zk  = T N . T ∗ N = inf   k

Proof of Proposition 7.S. Take any operator T ∈ B[X ] on a Hilbert space X . First we need the following four auxiliary results.

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7. Norms on Tensor Products

Claim 1.

|T |x = T x for every x ∈ X .

Proof of Claim 1 . |T |2 x ; x = T ∗ T x ; x = T x ; T x for every x ∈ X .

 

Claim 2. If T is compact, then there exist an orthonormal basis {eγ } for X and a family of nonnegative numbers {μγ } such that  μγ x ; eγ eγ for every x ∈ X . |T |x = γ

Proof of Claim 2 . The operator |T | ∈ B[X ] is nonnegative (thus self-adjoint and so normal) and compact. (In fact, as the class of compact operators from B[X ] is an ideal of B[X ], the nonnegative square root |T | of the nonnegative compact |T |2 is compact because |T |2 = T ∗ T is compact — see, e.g., [52, Problem 5.62].) Observe from Claim 1 that N (|T |) = N (T ). Then by the Spectral Theorem there is an orthonormal basis {ek } for the separable Hilbert space of |T | associated with positive eigenvalH = N (T )⊥ consisting of eigenvectors  ues {μk } of |T | such that |T |u = k μk u ; ek ek for every u ∈ H (see, e.g., [52, Corollary 6.44]). Since X = H ⊕ N with N = N (T ), there is an orthonormal basis {eγ } = {ek } ∪ {eδ } for X , where {eδ } is any orthonormal basis for the (not necessarily separable) Hilbert space N where |T |v = 0 for every v ∈ N , so that the above expansion on H actually describes |T |x for all x = u ⊕ v in   X = H ⊕ N with |T |eδ = 0, T ek = μk ek , and μγ = 0 if γ = k. Claim 3. An operator T lies in B1 [X ] if and only if there are unit X -valued sequences {zk } and{yk } (i.e., zk  = yk  = 1) and  a summable scalar se|α | < ∞) such that T x = quence {αk } (i.e., k k k αk x ; zk yk for every  x in X . Moreover, T 1 = inf k |αk ], where the infimum is taken over all summable scalar sequences for which the above expression for T x holds. Proof of Claim 3 . If T lies in B1 [X ], then T is compact. Consider Claim 2 and the setup in its proof. Take the direct sums T = T |H ⊕ O and |T | = |T ||H ⊕ O on X = H ⊕ N , where T |H ∈ B[H] is injective. Thus T |H has a polar decomposition T |H = V |T ||H where V ∈ B[H] is an isometry (see, e.g., [52, Corollary 5.90]). Set hk = V ek so that {hk } is an H-valued orthonormal sequence. Then   |T |x = μk x ; ek ek and so T x = μk x ; ek hk for every x ∈ X , k

k

since v ; ek  = 0 and hence x ; ek  = u ; ek  for all k and every x = u + v in X with u ∈ H and v ∈ N . Moreover, with μγ = 0 if γ = k,     μγ = μγ eγ ; eδ eδ = |T |eγ ; eγ  = T 1 < ∞. γ γ δ γ  Therefore T ∈ B1 [X ] implies T x = k μk x ; ek hk for every x ∈ X , where {ek } and {hk } are unit X -valued sequences and {μk } is a summable scalar sequence. Conversely, as X = N (T )⊥ ⊕ N (T ) we may regard the action of T = T |H ⊕ O and |T | = |T ||H ⊕ O only on H = N (T )⊥ . So for notational simplicity, write T and |T | for the injective operators T |H and |T ||H . Suppose  Tx = αk x ; zk yk for every x ∈ H, k

7.4 Additional Propositions

177

for some H-valued sequences {zk } and {yk } and for some scalar sequence {αk } with zk  = yk  = 1 and k |αk | < ∞. Thus (by polar decomposition)  |T |x = V ∗ T x = αk x ; zk wk for every x ∈ H, k

∗

where wk = V yk so that wk  ≤ 1. For any orthonormal basis {ej } for H,       |T |ej ; ej  = αk ej ; zk wk ; ej = αk ej ; zk wk ; ej  j j k j k        ≤ |αk |  ej ; zk wk ; ej  = |αk | |wk ; zk | ≤ |αk | < ∞. k j k k  So T lies in B1 [X ]. Moreover, since T x = k αk x ; zk yk for every x ∈ H,      μk = T 1 = |T |ek ; ek  ≤ |T |ek  = T ek  ≤ |αk |, k k k k k  for every orthonormal basis {ek } for H. Therefore T 1 = min k |αk | where the minimum is taken over all scalar sequences {αk } satisfying a (nuclear) representation of T as above.   Claim4. Take an arbitrary orthonormal basis {eγ } for X . If T ∈ B[X ] is such that γ T eγ 2 < ∞, then for every orthonormal basis {hγ } for X ,   T eγ 2 = T ∗ hγ 2 . γ

γ

Proof of Claim 4 . For arbitrary orthonormal bases {eγ } and {hγ },    |T eγ ; hδ |2 = |hδ ; T eγ |2 = |T ∗ hδ ; eγ |2 T eγ 2 = δ

δ

δ

so that       T eγ 2 = |T eγ ; hδ |2 = |T ∗ hδ ; eγ |2 = T ∗ hδ 2 γ

γ

δ

δ

γ

δ

whenever any of the families {T eγ 2 } or {T ∗ hδ 2 } is summable.

 

(a) By the definition of nuclear operator on a Hilbert space and nuclear norm, B1 [X ] = BN [X ]

and

 · 1 =  · N .

Indeed, the expression for T x in Claim 3 is precisely a nuclear representation for T. So with the infimum taken over all nuclear representations of T ∈ BN [X ], we get (for arbitrary unit sequences {zk }, {y k } and summable scalar sequence {αk }) T N = inf k |αk |zk yk  = inf k |αk | = T 1 , thus proving (a). (b) For any orthonormal basis {eγ } for X we get, by Claim 1,    |T |eγ ; eγ  ≤ |T |eγ  = T eγ . γ γ γ  So if γ T eγ  < ∞, then T ∈ B1 [X ]. Conversely, since trace-class operators are compact, Claim 2 ensures the existence of an orthonormal basis {eγ } for X and a sequence of nonnegative numbers {μγ } such that  |T |x = μγ x ; eγ eγ for every x ∈ X , γ

178

7. Norms on Tensor Products

where |T |eγ = μγ eγ (cf. proof of Claim 2). Then    T eγ  = |T |eγ  = μγ γ

γ

γ

by Claim 1 again. If T lies in B1 [X ], then   T 1 = |T |eγ ; eγ  = μγ < ∞. γ

Therefore if T ∈ B1 [X ], then



γ

γ

T eγ  < ∞. This concludes the proof of (b).

(c) Every T ∈ B1 [X ] can be factored as T = AB where A, B ∈ B[X ] are such that |A|2 and |B|2 lie in B1 [X  5.66(e)]). 2Equivalently, ] (see, e.g.,2 [52, Problem A, B ∈ B[X ] are such that γ |A|eγ  < ∞ and γ |B|eγ  < ∞ for an arbitrary orthonormal basis {eγ } for X . Since 2|T eγ ; eγ | = 2|ABeγ ; eγ | = 2|Beγ ; A∗ eγ | ≤ 2Beγ A∗ eγ  ≤ Beγ 2 + A∗ eγ 2, then by Claims 1 and 4      |T eγ ; eγ | ≤ Beγ 2 + A∗ eγ 2 = Beγ 2 + Aeγ 2 2 γ γ γ γ γ   = |B|eγ 2 + |A|eγ 2 < ∞. γ γ  Thus if T lies in B1 [X ], then γ |T eγ ; eγ | < ∞, concluding the proof of (c). (d) If S ∈ B[X ] is a unilateral shift of multiplicity one on a separable Hilbert space X (where orthonormal bases are countable), then (by definition) S shifts an orthonormal basis for X , which means Sek = ek+1 for each positive integer k for some orthonormal basis {ek } for X . Observe that Shk ; hk  = 0 for every orthonormal basis {hk } for X . Indeed, take any orthonormal  basis {hk } for X and the Fourier expansion of hk in terms of {ek }, say hk = j hk ; ej ej . So Shk = j hk ; ej Sej . Hence     hk ; ej ej+1 ; hk ; ei ei = hk ; ej hk ; ei ej+1 ; ei  Shk ; hk  = j i i,j   hk ; ej hk ; ej+1  = ej+1 ; hk ej ; hk  = ej+1 ; ej  = 0, = j

j

by  considering the Fourier expansion of each ek in terms of {hk }. Therefore k |Shk ; hk | = 0 for every orthonormal basis {hk }. However, as S is an isometry, S ∗ S = I and so |S| = I, the identity on X . Thus S ∈ B1 [X ] since n k=1 |S|hk ; hk  = n for every orthonormal basis {hk } for X , thus proving (d). This completes the proof of Proposition 7.S. (More along this line in [58].)   For a notion of trace beyond Hilbert spaces see, e.g., [26, 44, 75].

Suggested Readings Defant and Floret [15] Diestel, Fourie and Swart [17]

Ryan [83] Schatten [85]

8 Operator Norms

If X , Y, and Z are normed spaces over the same scalar field, and if T lies in B[X , Y] and S lies in B[Y, Z], then S T lies in B[X , Z] and S T  ≤ S T . The above inequality is a crucial property shared by the induced uniform norm of bounded linear transformations, referred to as the operator norm property. For X = Y = Z this property ensures that B[X ] is a normed algebra, where the composition S ◦ T stands for the product S T of S and T in the algebra. This chapter deals with a counterpart of that inequality for the case of tensor products rather than ordinary products. When is it the case that A ⊗ B ≤ A B ?

8.1 Uniform Crossnorms Each reasonable crossnorm in Chapter 7 has been defined for a given pair of normed spaces. From now on we consider a class of reasonable crossnorms that, among other attributes, are defined for every pair of normed spaces. Throughout the chapter, X , Y, V, W stand for normed spaces (obviously, over the same field). Take the tensor product spaces X ⊗ Y and V ⊗ W, let A ∈ B[X , V] and B ∈ B[Y, W] be bounded linear transformations, and consider the linear tensor product transformation A ⊗ B ∈ L[X ⊗ Y, V ⊗ W] as in Definition 3.16 and Theorem 3.17. Remark 8.1. Bounded Tensor Product Transformation. Let the tensor product spaces X ⊗ Y and V ⊗ W be equipped with reasonable crossnorms  · α and  · β , respectively, so that X ⊗α Y = (X ⊗ Y,  · α ) and V ⊗β W = (V ⊗ W,  · β ). For arbitrary bounded linear transformations A ∈ B[X , V] and B ∈ B[Y, W], consider their tensor product A ⊗ B ∈ L[X ⊗α Y, V ⊗β W]. If A ⊗ B is bounded (i.e., if A ⊗ B ∈ B[X ⊗α Y, V ⊗β W]), then © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Kubrusly, Bilinear Maps and Tensor Products in Operator Theory, Universitext, https://doi.org/10.1007/978-3-031-34093-2 8

179

180

8. Operator Norms

(a)

(A ⊗ B)V⊗βW ≤ A ⊗ BX ⊗α Y for every  ∈ X ⊗ Y,

(b)

A B ≤ A ⊗ B.

Indeed, as happens with bounded linear operators between normed spaces, (a) A ⊗ B = inf{c > 0 : (A ⊗ B)V⊗βW ≤ cX ⊗α Y ,  ∈ X ⊗α Y}, and (b) A ⊗ B = supα =1 (A ⊗ B)β ≥ supx⊗yα =1 (A ⊗ B)(x ⊗ y)β = supxy=1 Ax By ≥ supx=1 Ax supy=1 By = A B. (We have already used this argument in the proof of Theorem 7.8.) Let  · α be a reasonable crossnorm acting on the tensor product space X ⊗ Y for every pair of normed spaces X and Y. So for arbitrary normed spaces X , Y, V, W, it is a reasonable crossnorm on both tensor product spaces X ⊗ Y and V ⊗ W. Take bounded linear transformations A ∈ B[X , V] and B ∈ B[Y, W] and consider their tensor product A ⊗ B : X ⊗α Y → V ⊗α W, which is linear, acting between the crossnormed spaces X ⊗α Y and V ⊗α W. Definition 8.2. (a) A uniform crossnorm (plain uniform crossnorm)  · α is (a1) a reasonable crossnorm acting on every tensor product space such that (a2) for arbitrary normed spaces X , Y, V, W, and for every pair of bounded linear transformations A ∈ B[X , V] and B ∈ B[Y, W], the tensor product A ⊗ B is bounded — that is, A ⊗ B ∈ B[X ⊗α Y, V ⊗α W] — and A ⊗ B ≤ A B. This is the standard definition. It will be helpful to define two particular cases. (b) A finite-dimensional uniform crossnorm (or a uniform crossnorm on tensor product spaces of finite-dimensional normed spaces) is a reasonable crossnorm on the class of all tensor product spaces of finite-dimensional normed spaces, which acts on all such spaces as a uniform crossnorm. In other words, such that (a1 ) and (a2 ) above hold for all finite-dimensional normed spaces X , Y, V, W. Tautologically, (a) implies (b). A useful intermediate notion reads as follows. (c) A uniform crossnorm on finite-dimensional spaces is a reasonable crossnorm on every tensor product space (of arbitrary normed spaces) acting as a uniform crossnorm for bounded linear transformations defined on finitedimensional normed spaces. In other words,  · α is defined as a reasonable crossnorm on every tensor product space (of arbitrary normed spaces), but (a2 ) above holds for finite-dimensional normed spaces X , Y and for arbitrary normed spaces V, W (i.e., (a2 ) holds for bounded linear transformations with arbitrary finite-dimensional domains, with no restriction on codomains). The difference between (b) and (c) above is this. In (c) it is assumed that (1) the reasonable crossnorm  · α is defined on every tensor product space (of arbitrary normed spaces), (2) the tensor product transformation A ⊗ B lies in B[X ⊗α Y, V ⊗α W] for every A ∈ B[X , V] and B ∈ B[Y, W] if X and Y are finite-dimensional and, in this case, (3) the norm inequality in (a2 ) holds true.

8.1 Uniform Crossnorms

181

Note that R = R(A) ⊆ V and S = R(B) ⊆ W are finite-dimensional spaces (by the rank and nullity identity in Proposition 1.E), and so R ⊗α S is a finitedimensional (algebraic) linear manifold of (the possibly infinite-dimensional) V ⊗α W. Thus we may regard the bounded linear transformations A ∈ B[X, R], B ∈ B[Y, S], and A ⊗ B ∈ B[X ⊗α Y, R ⊗α S], as operating in a finite-dimensional setup. In (b) a similar scheme is assumed, but it is restricted to a further condition where V and W are required to be finite-dimensional, and  · α is not assumed to act on infinite-dimensional tensor product spaces. Theorem 8.3. Let  · α be an assignment of a reasonable crossnorm to every tensor product space. Let X , Y, V, W be arbitrary normed spaces. (a) The norm  · α is a uniform crossnorm of one of the types defined in Definition 8.2 if and only if A ⊗ B = A B whenever A ⊗ B ∈ B[X ⊗α Y, V ⊗α W] for A ∈ B[X , V] and B ∈ B[Y, W]. (b) If M and N are linear manifolds of arbitrary normed spaces X and Y, respectively, and if  · α is a uniform crossnorm (or a uniform crossnorm on finite-dimensional spaces), then X ⊗α Y ≤ M⊗α N

for every

 ∈ M ⊗ N.

Note: in case of a uniform crossnorm on finite-dimensional spaces as in Definition 8.2(c), M and N are assumed to be finite-dimensional . (c) If  · α is a uniform crossnorm, then B[X , V] ⊗ B[Y, W] ⊆ B[X ⊗α Y, V ⊗α W]. Note: this extends a condition satisfied for every reasonable crossnorm, X ∗ ⊗ Y ∗ ⊆ B[X ⊗α Y, F ⊗α F ] ∼ = (X ⊗α Y)∗ . Proof. (a) The identity in item (a) is straightforward from Remark 8.1(b) and Definition 8.2 (regarding the nature — finite-dimensional or not — of X , Y, V, W according to each of the options (a), (b) or (c) in Definition 8.2). (b) Suppose  · α is a uniform crossnorm or a uniform crossnorm on finite-dimensional spaces as in Definition 8.2(a,c). Let M and N be (finite-dimensional, in the case of Definition 8.2(c)) linear manifolds of X and Y. Take the natural isometric embeddings IM : M → X and IN : N → Y. Since  · α is uniform, IM ∈ B[M, X ], IN ∈ B[N, Y], IM  = IN  = 1, it follows that IM ⊗ IN lies in B[M ⊗α N , X ⊗α Y] with IM ⊗ IN  = IM IN  = 1. So for  ∈ M ⊗ N , X ⊗α Y = (IM ⊗ IN )X ⊗α Y ≤ IM ⊗ IN M ⊗α N = M⊗α N . (c)  An element in the tensor product space B[X , V] ⊗ B[Y, W] is a finite sum i Ai ⊗ Bi of single tensors with Ai ∈ B[X , V] and Bi ∈ B[Y, W]. Since  · α

182

8. Operator Norms

is a uniform crossnorm, each single tensor Ai ⊗ Bi lies in B[X ⊗α Y, V ⊗α W] by Definition 8.2(a). So i Ai ⊗ Bi lies in the linear space B[X ⊗α Y, V ⊗α W], leading to the first claimed inclusion (compare with Theorem 3.17 (d)). For the particular case, set V = W = F and note that F ⊗α F ∼ = F for every reasonable crossnorm  · α on the one-dimensional tensor product space F ⊗ F (i.e., such that a ⊗ bα = a · bα ≤ |a| |b| for every a, b ∈ F ). Thus B[X ⊗α Y, F ⊗α F ] ∼ = B[(X ⊗ Y,  · α ), (F , | · |)] = (X ⊗α Y)∗ (cf. Remark 3.18).   α Y and V ⊗  α W be the completions of X ⊗α Y and V ⊗α W (where Let X ⊗  · α is a reasonable crossnorm on X ⊗Y and on V ⊗ W). Suppose A ⊗ B  lies in B[X ⊗α Y, V ⊗α W] for some A ∈ B[X , V] and B ∈ B[Y, W]. Let A ⊗B  α W] be the extension over completion of A ⊗ B. Recall that  α Y, V ⊗ in B[X ⊗  B = A ⊗ B, and hence  · α is a uniform crossnorm if and only if A ⊗  B = A B for every A and B by Theorem 8.3(a). A ⊗ Theorem 8.4. Projective and injective norms are uniform crossnorms. Proof. Let X , Y, V, W be normed spaces. Suppose a reasonable crossnorm  · α acts on both tensor product spaces X ⊗ Y and V ⊗ W. Take A ∈ B[X , V] and B ∈ B[Y, W]. Consider their tensor product A ⊗B in L[X ⊗α Y, V ⊗α W]. Let  = i xi ⊗ yi be any (finite) representation of an arbitrary  ∈ X ⊗α Y.  (a) Thus for every representation i xi ⊗ yi of  ∈ X ⊗ Y,      (A ⊗ B)α =  Axi ⊗ Byi  ≤ Axi ⊗ Byi α i i α   = Axi  Byi  ≤ A B xi  yi . i

i

By definition, the injective  · ∨ and projective  · ∧ norms can be assigned to every tensor product of normed spaces. Then, to begin with, set  · α =  · ∧.  In this case ∧ = inf i xi  yi , and so by the above inequality we get  xi  yi  = A B ∧. (A ⊗ B)∧ ≤ A B inf i

Therefore A ⊗ B ∈ L[X ⊗∧ Y, V ⊗∧ W] is bounded and A ⊗ B ≤ A B. Thus  · ∧ is a uniform crossnorm by Definition 8.2. (b) Set  · α =  · ∨. Consider the adjoints A∗∈ B[V ∗, X ∗ ] and B ∗∈ B[W ∗, Y ∗ ]. g ∈ W ∗ :  g  ≤ 1}. Thus Take the balls BV ∗ = {f ∈ V ∗ : f ≤ 1} and BW ∗ = {         (A ⊗ B)∨ =  Axi ⊗ Byi  = sup f(Axi ) g(Byi )  i

=

sup ˜∈BW ∗ f˜∈BV ∗ , g

=

sup ˜∈BW ∗ f˜∈BV ∗ , g

∨

f˜∈B

∗, g ˜∈B

∗

i

V W     ∗ ∗ (A f )(xi ) (B g)(yi ) 

i

  ∗    A  (xi ) B ∗∗ g (yi ). A∗  B ∗  f ∗ A  B  i

A∗ f B∗ g  ∗ ∗ For each f ∈ V ∗ and g ∈ W ∗ set f  = A and g  = B ∗ ∈ X ∗  ∈ Y , and take ∗ the normalized images of the unit balls BV ∗ and BW ∗ under A and B ∗,

8.1 Uniform Crossnorms BX ∗ = BY ∗ =

∗ 1 ∗ A∗  A (BV ) ∗ 1 ∗ B ∗  B (BW )

= f =

= g =

A∗ f A∗ 

 ∈ X ∗ : f ∈ V ∗, f ≤ 1 ,

B∗ g  B ∗ 

 ∈ Y ∗ : g ∈ W ∗,  g ≤ 1 .

183

Then (as f ∈ BV ∗ ⇔ f  ∈ BX ∗ and g ∈ BW ∗ ⇔ g  ∈ BY ∗ )     sup f  (xi ) g  (yi ). (A ⊗ B)∨ = A∗  B ∗   f  ∈BX ∗ , g  ∈BY ∗

i

Take the balls BX ∗ = {f ∈ X ∗ : f  ≤ 1} and BY ∗ = {g ∈ Y ∗ : g ≤ 1} so that BX ∗ ⊆ BX ∗ and BY ∗ ⊆ BY ∗ (as f ≤ 1 ⇔ f   ≤ 1 and  g  ≤ 1 ⇔ g   ≤ 1). ∗ ∗ Hence (since A  = A and B  = B) we get     sup f (xi ) g(yi ) = A B ∨. (A ⊗ B)∨ ≤ A B  f ∈BX ∗ , g∈BY ∗

i

Therefore A ⊗ B ∈ L[X ⊗∨ Y, V ⊗∨ W] is bounded and A ⊗ B ≤ A B.  Thus  · ∨ is a uniform crossnorm by Definition 8.2. Corollary 8.5. If A ∈ B[X , V] and B ∈ B[Y, W] are quotient transforma W] is a quotient transformation.  B ∈ B[X ⊗  Y, V ⊗ tions, then A ⊗ ∧ ∧ Proof. Let X , Y, V, W be normed spaces. Take A ∈ B[X , V] and B ∈ B[X , W]. Equip the tensor product spaces X ⊗ Y and V ⊗ W with the projective norm. Thus A ⊗B ∈ B[X ⊗∧ Y, V ⊗∧ W] and A ⊗B = A B by Theorems 8.3 and 8.4. Suppose A : X → V and B : Y → W are quotient transformations, and take an arbitrary ε > 0. Then, according to Definition 5.12, (i) A and B are surjective and hence A ⊗ B is surjective by Theorem 3.19(e), (ii) for every 0 = v ∈ V and every 0 = w ∈ W there are x ∈ A−1 ({v}) ⊆ X and y ∈ B −1 ({w}) ⊆ Y such that x2 < (1 + ε)v2 and y2 < (1 + ε)w2 , (iii) A = B = 1 and so A ⊗B = 1 by Theorems 5.13(e), 8.3 and 8.4. Take an arbitrary 0 = G ∈ V ⊗∧ W. Consider the definition of G∧ and let  ∈ V ⊗∧ W be a representation of G with nonzero single tensors i vi ⊗ wi such that i vi  wi  < G∧ + ε. By (i) and (ii), for each 0 = vi ∈ V and 0 = wi ∈ W there are xi ∈ A−1 ({vi }) ⊆ X and yi ∈ B −1 ({wi }) ⊆ Y such that (1) xi 2 ≤ (1 + ε)vi 2 and yi 2 ≤ (1 + ε)wi 2 .  Associated with V ⊗∧ W, set  = i xi ⊗ yi ∈ X ⊗ Y so that  such a G ∈ (A ⊗ B) = i Axi ⊗ Byi = i vi ⊗ wi = G by Definition 3.16. Then  (2)  = i xi ⊗ yi ∈ (A ⊗ B)−1 ({G}) ⊆ (X ⊗∧ Y). Moreover, by property (iii), (3) G∧ = (A ⊗ B)∧ ≤ ∧. Also, by property (1),

184

8. Operator Norms

      ∧ =  xi ⊗ yi  ≤ xi yi  ≤ (1 + ε) vi wi  < (1 + ε)(G∧ + ε). i

∧

i

i

Thus properties (1), (2) and (3) say that for an arbitrary nonzero G ∈ V ⊗∧ W and an arbitrary ε > 0 there exists  = G,ε ∈ (A ⊗ B)−1 ({G)} for which G∧ ≤ ∧ ≤ (1 + ε) (G∧ + ε). Hence G∧ =

inf

∈(A⊗B)−1 ({G})

∧.

That is, A ⊗B in B[X ⊗∧ Y, V ⊗∧ W] is a quotient transformation (Definition  W]  in B[X ⊗  Y, V ⊗ 5.12). Thus Theorem 5.13(j) says that its extension A ⊗B ∧ ∧   over the completions X ⊗∧ Y and V ⊗∧ W of the tensor products X ⊗∧ Y and V ⊗∧ W is a quotient transformation as well.  Remark 8.6. Characterizing Projective and Injective Uniform Crossnorms. Suppose M and N are subspaces of X and Y, respectively. (a) By Theorem 5.8, M and N must be closed so that X /M and Y/N are naturally normed. By Theorem 5.13(f), the quotient maps πX ∈ B[X , X /M] and πY ∈ B[Y, Y/N ] (which are contractions with norm one) are quotient transformations for every pair of proper subspaces M and N of X and Y. Then, by setting V = X /M and W = Y/N in Corollary 8.5, it follows that πX ⊗∧ πY in  πY in B[X ⊗∧ Y, X /M ⊗∧ Y/N ] is a quotient transformation, and thus is πX ⊗ ∧   B[X ⊗∧ Y, X /M ⊗∧ Y/N ]. So Corollary 7.11 and Theorem 7.10 (whose proofs are based on Theorem 7.9) can be viewed as consequences of Corollary 8.5. A projective uniform crossnorm is a uniform crossnorm for which Corollary 8.5 holds (and so Corollary 7.11 and Theorem 7.10 hold). That is, a projective uniform crossnorm is a uniform crossnorm  · α for which A ⊗ B ∈ B[X ⊗α Y, V ⊗α W] is a quotient transformation whenever A ∈ B[X , V] and B ∈ B[Y, W] are. (b) Let M and N be subspaces of X and Y and take an arbitrary  in M ⊗ N. By Theorem 8.3(b), M⊗α N ≤ X ⊗α Y for every uniform crossnorm  · α. By Theorem 8.4, the projective norm  · ∧ is uniform, and so M ⊗∧ N ≤ X ⊗∧ Y but, as observed in Remark 7.5, the inequality may be strict. In other words, although M ⊗ N is a regular manifold of the linear space X ⊗ Y, when both are equipped with the projective norm  · ∧ it may happen that M ⊗∧ N < X ⊗∧ Y for some  in M ⊗∧ N and, consequently, M ⊗∧ N may not be a linear manifold of the normed space X ⊗∧ Y. For the injective norm  · ∨, however, Theorem 7.19 says that M ⊗∨ N = X ⊗∨ Y for every  in M ⊗∨ N , and so M ⊗∨ N is a linear manifold of the normed space X ⊗∨ Y. An injective uniform crossnorm is a uniform crossnorm for which Theorem 7.19 holds. That is, an injective uniform crossnorm is a uniform crossnorm  · α for which M ⊗α N is a linear manifold of the normed space X ⊗α Y whenever M and N are subspaces of X and Y. Equivalently, for which M⊗α N = X ⊗α Y for every  in M ⊗α N .

8.2 Tensor Norms

185

8.2 Tensor Norms The notion of uniform crossnorms allows us to proceed towards a theory of normed tensor products of bounded linear operators. Indeed, uniform crossnorms apply to tensor product spaces of every pair of normed spaces. Moreover, a tensor product of bounded linear operators acting between tensor product spaces equipped with a uniform crossnorm is bounded and, in addition, its norm satisfies the operator norm property (regarding tensor products). There is, however, an additional property that is still desirable: finite generation. Definition 8.7. Let  · α be a reasonable crossnorm defined on every tensor product space X ⊗ Y for arbitrary normed spaces X and Y. Let X ⊗α Y stand for the normed space (X ⊗ Y,  · α ), as usual. (a) The reasonable crossnorm  · α is said to be finitely generated if, for every pair {X , Y} of normed spaces and every  ∈ X ⊗ Y, X ⊗α Y =

inf

dim M