Normed Linear Spaces [3 ed.] 9780444108227, 044410822X, 9783540061489, 3540061487

Table of contents :
Foreword to the First Edition
Foreword to the Third Edition
Contents
Chapter I. Linear Spaces
§ 1. Linear Spaces and Linear Dependence
§ 2. Linear Functions and Conjugate Spaces
§ 3. The Hahn-Banach Extension Theorem
§ 4. Linear Topological Spaces
§ 5. Conjugate Spaces
§ 6. Cones, Wedges, Order Relations
Chapter II. Normed Linear Spaces
§ 1. Elementary Definitions and Properties
§ 2. Examples of Normed Spaces; Constructions of New Spaces from Old
§ 3. Category Proofs
§ 4. Geometry and Approximation
§ 5. Comparison of Topologies in a Normed Space
Chapter III. Completeness, Compactness, and Reflexivity
§ 1. Completeness in a Linear Topological Space
§ 2. Compactness
§ 3. Completely Continuous Linear Operators
§ 4. Reflexivity
§ 5. Weak Compactness and Structure in Normed Spaces
Chapter IV. Unconditional Convergence and Bases
§ 1. Series and Unconditional Convergence
§ 2. Tensor Products of Locally Convex Spaces
§ 3. Schauder Bases in Separable Spaces
§ 4. Unconditional Bases
Chapter V. Compact Convex Sets and Continuous Function Spaces
§ 1. Extreme Points of Compact Convex Sets
§ 2. Fixed-point Theorems
§ 3. Some Properties of Continuous Function Spaces
§ 4. Characterizations of Continuous Function Spaces among Banach Spaces
Chapter VI. Norm and Order
§ 1. Vector Lattices and Normed Lattices
§ 2. Linear Sublattices of Continuous Function Spaces
§ 3. Monotone Projections and Extensions
§ 4. Special Properties of (AL)-Spaces
Chapter VII. Metric Geometry in Normecl Spaces
§ 1. Isometry and the Linear Structure
§ 2. Rotundity and Smoothness
§ 3. Characterizations of Inner-Product Spaces
§ 4. Isomorphisms to Improve the Norm
A. Rotundity, smoothness, and convex functions
B. Superreflexive spaces
Chapter VIII. Reader's Guide
A: To 1956
B: To 1972
Bibliography
A・B
C
D
E・F・G
H
J・K
L・M
N・O・P
R
S
T・U・V・W・Y・Z
Index of Citations
Index of Symbols
ConstantsJrom Set Theory and Logic
Non-alphabetical Symbols Jor Mathematical Operations
Symbols which are Names or Abbreviations of Names
Symbols used as Labels for Properties or Conditions
Subject Index

Citation preview

Mahlon M. Day

Normed Linear Spaces

Third Edition

Springer-Verlag Berlin Heidelberg New York

1973

Mahlon M. Day University of Illinois, Urbana, Illinois, U.SA.

AMS Subject Classifications (1970): Primary 46Bxx· Secondary 46Axx

ISBN 3-540-06148-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-06148-7 Springer-Verlag New York Heidelberg Berlin

ISBN 3·540-02811-0 Second edition Springer-Verlag Berlin Heidelberg New York ISBN 0-387-02811-0 Second edition Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher © by Springer-Verlag Berlin Heidelberg 1958, 1962, 1973 Library of Congress Catalog Card Number 72-96940 Printed in Germany Monophoto typesetting and offset printing Zechnersche Buchdruckerei, Speyer Bookbinding Konrad TriItsch, Wurzburg

Foreword to the First Edition

This book contains a compressed introduction to the study of normed linear spaces and to that part of the theory of linear topological spaces without which the main discussion could not well proceed. Definitions of many terms which are required in passing can be found in the alphabetical index. Symbols which are used throughout all, or a significant part, of this book are indexed on page 132*. Each reference to the bibliography is made by means of the author's name, supplemented when necessary by a number in square brackets. The bibliography does not completely cover the available literature, even the most recent; each PFlper in it is the subject of a specific reference at some point in the text. 1 The writer takes this opportunity to express thanks to the University of Illinois, the National Science Foundation, and the University of Washington, each of which has contributed in some degree to the cultural, financial, or physical support of the writer, and to Mr. R .. R. Phelps, who eradicated many of the errors with which the manuscript was infested. Urbana, Illinois (USA), September 1957

Mahlon M. Day

Foreword to the Third Edition The major changes in this edition are: Corrections and additions to II, § 5, and III, § 1, on hw* and ew* topologies; enlargement of III, § 2, on weak compactness, V, § 1, on extreme and exposed points, V, § 2, on fixed points, and VII, § 2, on rotundity and smoothness; added sections III, § 5, on weak compactness and structure of normed spaces, and VII, § 4, on isomorphism to improve the norm, and Index of Citations. Urbana, September 1972

*

In the 3rd edition page 201.

M.M.D.

Contents

Chapter I. Linear Spaces . . . . . . . . . . § 1. Linear Spaces and Linear Dependence . § 2. Linear Functions and Conjugate Spaces. § 3. The Hahn-Banach Extension Theorem § 4. Linear Topological Spaces. . . . § 5. Conjugate Spaces . . . . . . . § 6. Cones, Wedges, Order Relations.

1 1 4 9 12 18 22

Chapter II. Normed Linear Spaces. . . § 1. Elementary Definitions and Properties § 2. Examples of Normed Spaces; Constructions of New Spaces from Old. . . . . . . . . . § 3. Category Proofs. . . . . . . . . . . . . . § 4. Geometry and Approximation. . . . . . . . § 5. Comparison of Topologies in a Normed Space.

27 27

Chapter III. Completeness, Compactness, and Reflexivity § 1. Completeness in a Linear Topological Space. § 2. Compactness . . . . . . . . . . . . § 3. Completely Continuous Linear Operators. . § 4. Reflexivity . . . . . . . . . . . . . . . § 5. Weak Compactness and Structure in Normed Spaces

53 53 57 65 69 72

Chapter IV. Unconditional Convergence and Bases.

78

§ 1. § 2. § 3. § 4.

Series and Unconditional Convergence . . Tensor Products of Locally Convex Spaces Schauder Bases in Separable Spaces. Unconditional Bases . . . . . . . . . .

32 38 43 45

78 83 87 95

Chapter V. Compact Convex Sets and Continuous Function Spaces 101 § 1. Extreme Points of Compact Convex Sets . . 101 § 2. Fixed-point Theorems . . . . . . . . . . . . . . . . 106

VIII

Contents

§ 3. Some Properties of Continuous Function Spaces . . . 113 § 4. Characterizations of Continuous Function Spaces among 116 Banach Spaces. . . . Chapter VI. Norm and Order § 1. Vector Lattices and Normed Lattices § 2. Linear Sublattices of Continuous Function Spaces § 3. Monotone Projections and Extensions . . § 4. Special Properties of (AL)-Spaces . . . .

126 126 131 135 137

Chapter VII. Metric Geometry in Normed Spaces

142

§ 1. Isometry and the Linear Structure . . . . § 2. Rotundity and Smoothness . . . . . . .

142 144 151 159 159 168

§ 3. Characterizations of Inner-Product Spaces § 4. Isomorphisms to Improve the Norm . . . A. Rotundity, smoothness, and convex functions. B. Superreflexive spaces Chapter VIII. Reader's Guide

175

Bibliography . .

184

Index of Citations

197

Index of Symbols

201

Subject Index . .

205

Chapter I. Linear Spaces

§ 1. Linear Spaces and Linear Dependence The axioms of a linear or vector space have been chosen to display some of the algebraic properties common to many classes of functions appearing frequently in analysis. Of these examples there is no doubt that the most fundamental, and earliest, examples are furnished by the n-dimensional Euclidean spaces and their vector algebras. Nearly as important, and the basic examples for most of this book, are many function spaces; for example, C[O, 1], the space of real-valued continuous functions on the closed unit interval, B V [0, 1], the space of functions of bounded variation on the same interval, I! [0, 1], the space of those Lebesgue measurable functions on the same interval which have summable plh powers, and A(D), the space of all complexvalued functions analytic in a domain D of the complex plane. Though all these examples have further noteworthy properties, all share a common algebraic pattern which is axiomatized as follows: (Banach, p. 26; Jacobson).

Definition 1. A linear space L over a field A is a set of elements satisfying the following conditions: (A) The set L is an Abelian group under an operation +; that is, + is defined from L x L into L such that, for every x, y, z in L, (a) x+y=y+x, (commutativity) (b) x+(y+z)=(x+y)+z, (associativity) (c) there is a w dependent on x and y such that x+w=y. (B) There is an operation defined from A x L into L, symbolized by juxtaposition, such that, for A, f1 in A and x, y in L, (d) (e) (1) (g)

A(X + y) = Ax + Ay , (A+f1)X=AX+f1X, A(f1 x) = (A f1) x ,

(distributivity) (distributivity)

1 x = x (where 1 is the identity element of the field).

Chapter I. Linear Spaces

2

In this and the next section any field not of characteristic 2 will do; in the rest of the book order and distance are important, so the real field R is used throughout, with remarks about the complex case when that field can be used instead. (1) If L is a linear space, then (a) there is a unique element 0 in L such that x+O=O+x=x and ,uO=Ox=O for all ,u in A and x in L; (b) ,ux=O if and only if ,u=O or x=O; (c) for each x in L there is a unique y in L such that x+y=y+x=O and (-1)x=y; (then for z,xinLdefine z-x=z+(-1)x and -x=O-x). (2) It can be shown by induction on the number of terms that the commutative, associative and distributive laws hold for arbitrarily large finite sets of elements; for example, L Xi' which is defined to be i::;n

Xl

+ (X2 + (... + x n)···), is independent of the order or grouping of terms

in the process of addition. Definition 2. A non-empty subset £, is called a linear subspace of L if £, is itself a linear space when the operations used in £, are those induced by the operations in L. If X =!= y, the line through x and y is the set {,ux+(1- ,u)y: ,uEA}. A non-empty subset E of L isjlat if with each pair x =!= y of its points E also contains the line through x and y. (3) £, is a linear subspace of L if and only if for each x, y in £, and each A in A, x + y and Ax are in £'. Definition 3. If E, F ~ Land z E L, define E+F={x+y:XEE and YEF}. E+z={X+Z:XEE},

-E={-X:XEE},

E-z=E+(-z),

E-F=E+(-F).

(4) (a) E is flat if and only if for each x in E the set E - x is a linear subspace of L. (b) The intersection of any family of linear [flat] subsets of L is linear [either empty or flat J. (c) Hence each non-empty subset E of L is contained in a smallest linear [flat] subset of L, called the linear [jlat} hull of E.

Definition 4. If L is a linear space and Xl' ... , Xn are points of L, a point x is a linear combination of these Xi if there exist Al' ... , An in A such that x = L Ai Xi. A set of points E ~ L is called linearly independent i::;n

if E is not-0 or {O} and l no point of E is a linear combination of any finite subset of the other points of E. A vector basis (or Hamel basis) in L is a maximal linearly independent set.

(5) (a) The set of all linear combinations of all finite subsets of a set E in L is the linear hull of E. (b) E is linearly independent if and 1

0 is the empty set; {x} is the set containing the single element x.

§ 1. Linear Spaces and Linear Dependence

3

only if for Xl' ... , Xn distinct elements of E and ,11' ... , An in A the condition L Aixi=O implies that A1 =A2=···=An=0. i~n

Theorem 1. If E is a linearly independent set in L, then there is a vector basis B of L such that B"ii?,E. Proof. Let 6 be the set of all linearly independent subsets S of L such that E~S; let Sl~S2 mean that Sl"ii?,S2. Then if6 0 isa simply ordered subsystem of 6 and So is the union of all S in 6 0 , So is also. a linearly independent set; indeed, Xl' ... , Xn in So imply that there exist Si in 6 0 with Xi in Si. Since 6 0 is simply ordered by inclusion, all Xi belong to the largest Sj and are, therefore, linearly independent. Hence SoE 6 and is an upper bound for 6 0 . Zorn's lemma now applies to assert that E is contained in a maximal element B of 6. This B is the desired vector basis, for it is a linearly independent set and no linearly independent set is larger. Corollary 1. If Lo is a linear subspace of Land Bo is a vector basis for L o, then L has a vector basis B"ii?,Bo. (6) If B = {xs: s E S} is a vector basis in L, each X in L has a representation X= L AsXs' where (J is a finite subset of S. If X= L AsXs

=

L

SEa

SEUl

JlsX" then As=Jls for all s in (Jl (l(J2 and As=O for all other s

SE a 2

in (Jl and Jls=O for all other s in (J2. Hence each X=FO has a unique representation in which all coefficients are non-zero, and 0 has no representation in which any coefficient is non-zero. [Also see §2, (2c).] This property characterizes bases among subsets of L. Theorem 2. Any two vector bases Sand T of a linear space L have the same cardinal number. Proof. Symmetry of our assumptions and the Schroeder-Bernstein theorem on comparability of cardinals (Kelley, p. 28) show that it suffices to prove that S can be matched with a subset of T. Consider the transitively ordered system of functions ([> consisting of those functions cP such that (a) the domain Dep~S and the range Rep~ T. (b) cP is one-to-one between Dep and Rep. (c) Rep u (S\Dep) is a linearly independent set. Order ([> by: cP ~ cp' means that cp is an extension of cp'. Every simply ordered subsystem ([>0 of ([> has an upper bound CPo: Define Depo= U Dep and CPo(s) = cp(s) if sEDep and CPE([>o· This CPo epetPo

is defined and is in ([>; it is an upper bound for ([>0. By Zorn's lemma there is a maximal cp in ([>. We wish to show that Dep=S.

Chapter I. Linear Spaces

4

If not, then Rq> =!= T, for each s in the complement of Dq> is dependent on T but not on Rq>. If to is in T\Rq> , either to is linearly independent of Rq> U (S\Dq» or is dependent on it. In the former case, for arbitrary So in S\Dq> the extension cp' of cp for which cp' (so) = to has the properties (a), (b), and (c), so cp is not maximal. In the latter case, by (c) and (6)

to=

L

tER..

Att+

L

Ils S '

s$D..

where at least one Ils o is not zero, because to is independent of Rq>' If cp' is the extension of cp for which cp'(so)=t o, then cp' obviously satisfies (a) and (b); also Rq>' U (S\Dq>') is linearly independent, because otherwise to would depend on Rq> U (S\Dq>')' a possibility prevented by the choice of So, and again cp cannot be maximal. This shows that if cp is maximal in ([>, then Dq>=S; then the cardinal number of S is not greater than that of T. The Schroeder-Bernstein theorem completes the proof of the theorem. Definition 5. The cardinal number of a vector basis of L is called the dimension of L. The linear space with no element but 0 is the only linear space with an empty vector basis; it is the unique linear space of dimension O. (7) (a) If K is the complex field and if L is a vector space over K, then L is also a vector space, which we shall call L(r)' over the real field R. (b) The dimension of L(r) is twice that of L, for x and ix are linearly independent in L(r)'

§ 2. Linear Functions and Conjugate Spaces In this section again the nature of the field of scalars is unimportant as long as it is not of characteristic 2. Definition 1. If Land r. are linear spaces over the same field A, a function F (sometimes to be called an operator) from L into r. is called additive if F(x+y)=F(x)+F(y) for all x,y in L; homogeneous if F(A x) = AF(x) for all A in A and x in L; linear if both additive and homogeneous. A one-to-one linear F carrying L onto r. is an isomorphism of Land r.. (1) (a) Let B be a vector basis of L and for each b in B let Yb be a point of the linear space r.. Then there is a unique linear function F from L into r. such that F(b) = Yb for all bin B; precisely, using §1, (6),

F

(L Abb) = L AbYb' bEI1

bEI1

(b) If To is a linear function defined from a linear subspace Lo of L into a linear space r., there is an extension T of To defined from L into r..

§ 2. Linear Functions and Conjugate Spaces

5

(c) T is called idempotent if T Tx= Tx for all x in L. If Lo is a linear subspace of L, there is an idempotent linear function (a projection) P from L onto Lo. (d) There is an isomorphism between Land L if and only if these spaces have the same dimension. Linear extension problems are much simplified by the basis theorems. Lemma 1. Let L and ~ be linear spaces over A and let X be a subset of L, and let f be a function from X into L. Then there is a linear function F from L into r; such that F is an extension of f if and only if whenever a linear combination of elements of X vanishes, then the same linear combination of the corresponding values of f also vanishes; i.e., if L ..1.ixi=O then L..1.J(xi)=O. i

i

Proof. The necessity is an immediate consequence of the linearity of F. If the condition holds, define g at any point y= L..1.ixi in L o, i

the linear hull of X, by g(y)= L ..1.J(xJ If also y= L ..1.jxj, then j

j

L ..1.ixi- L AjXj=O so L ..1.J(x;)- L ..1.j!(xj)=O, and g(y) is deteri

j

i

j

mined by y, not by its representations in terms of X. This shows at once that g is linear on Lo; (1 b) asserts that g has a linear extension F.

Definition 2. (a) If L is a linear space, then L * , the conjugate space of L, is the set of all linear functions from L into the field A. (b) Let S be a non-empty set of indices and for each s in S let Ls be a linear space Ls be the set of all functions x on S such that X(S)E Ls over A. Let

n

SES

for all s in S; let L Ls be the subset of SES

n Ls consisting of those funcseS

tions x for which {s: x(s)=I=O} is fmite. Then these function spaces are linear spaces under the definitions (x+ y)(s) = x(s) + y(s)

and (A x)(s) = A(x (s))

for all x, y and all ..1.. They are called, respectively, the direct product and direct sum of the spaces Ls. (c) IJ is the special direct product in which all Ls = L. (2) (a) L* is a linear subspace of AL; hence L* is a linear space. (b) ( L Ls)* is isomorphic to (L.*). (c) If {xs: SES} is a basis for

n

SES

.eS

L and if for each s in S'/s is the unique element of L* such that fs(x s) = 1, fs(xs')=O if s'=I=s, then for each x in L, O'x= {s: fs(x) =l=O} is a finite subset of S and for every non-empty O'~O'x' X= Lfs(x)xs . sea

(d) If {xs: SES} is a basis in L, then L is isomorphic to L L., where SES

Chapter I. Linear Spaces

6

each Ls = A, and L# is isomorphic to AS. (e) If Xi' 1 ~ i ~ n, are linearly independent elements of L and if Ai' 1 ~ i ~ n, are in A, then there existsfin L# such that f(X;)=Ai, 1~i~n. Definition 3. A hyperplane H in L is a maximal flat proper subset of L, that is, H is flat, and if H' 2 H and H' is flat, then H' = H or H' = L. (3) (a) H is a hyperplane in L if and only if H is a translation X + Lo of a maximal linear proper subspace Lo of L. (b) If f E L #, if f is not 0, and if AEA, then {x:f(X)=A} is a hyperplane in L. (c) For each hyperplane H in L there is an f =F 0, f E L #, and a A in A such that H={XEL:f(x)=A}; H is linear if and only if A=O. (d) If the hyperplane H = {x:fdx)=Ad = {X:f2(X)=A 2}, then there exists ,u=FO in A such that fl = ,uf2 and Al = ,uA2· Definition 4. If Lo is a linear subspace of 1" define a vector structure on the factor space L/Lo of all translates, X + L o , of Lo as follows: If X and Yare translates of Lo define X + Y as in § 1, Def. 3 to be {x+ y: XEX and yE Y}; define AX to be {AX: XEX} if A =FO, OX =Lo. Let To be the function carrying X in L to X + Lo in L/ Lo . Theorem 1. L/Lo is a vector space and To is a linear function from L onto L/Lo . Proof. If XEX and YEY, then X=Tox=x+L o and Y=ToY =y+L o . Hence X+Y={x+y+u+v:u,vEL o} = {x+y+W:wELo} = (x+y)+L o = To(x+ y). Hence X + YEL/L o , and To is additive. Similarly X = X + L o , so Lo is the zero element of L/Lo. If A =F 0, then AX =ATOX=A{X+U: uELo} = {Ax+AU: uELo} = {Ax+v: VELo} = To (Ax), so AXEL/Lo and TO(AX)=ATo(x). If A=O, OX=L o = To (0) = To (0 x), so To is homogeneous. Associativity, distributivity, and so on, are easily checked. Next we improve the result of (2e).

all

Definition 5. A subset r of L# is called total over L if f(x)=O for in r implies that x = 0.

f

Theorem 2. Let r be a linear subspace of L # which is total over L and let Xi' i=1, ... , n, be linearly independent elements of L; then there exists elements fi' i = 1, ... , n, in r such that f;(xj) = (jij (Kronecker's delta) for i,j=1, ... ,n. Proof. To prove this by induction on n, begin with n = 1. If Xl is a linearly independent set, then Xl =FO; hence, by totality there is an f in r with f(x[)=FO; set fl = f/f(x l ). Assume the result true for n -1 and let Xl' ... ' xn be independent. Then there exist f;, ... ,f~-l such that f;(x) = (jij for i,j = 1, ... , n-1.

§ 2. Linear Functions and Conjugate Spaces

7

Let T map r into An by (Tf)j=f(xj), j=1, ... ,n. We wish to show T is onto An, so we suppose, for a contradiction, that Tf is linearly dependent on the Tfi, iO, then (rE)"=E"/r. (d) If E1 r;;;;,E 2 , then Ei~E2' (e) If K(E) = closed convex hull of E, then K(E)"=E"

§ 5. Conjugate Spaces

19

= K(E u {O})". (f) (E1 u E 2)" = Ef n E'2. (g) If E is symmetric or linear, so is E". These are many of the properties of neighborhoods of 0 in L*; to get topologies in L* we choose suitable families of sets in L. Definition 2. A family (f of subsets of an L TS L is called a topologizing family for L* if (a) each finite subset

O. For discussions of the operation of lineal closure see Klee [5] and Nikodym. (11) If W is a wedge in Land W" is the polar set in L*, then (a) W"= {f:fEL* and f(x)~O for all x in W}. (b) W" is a w*-closed wedge in L*. (12) Tukey showed that two closed convex disjoint sets in a reflexive Banach space can be separated by a closed hyperplane. Dieudonne [5] showed that this property fails in P(OJ). Klee [2] shows how local compactness is needed in the separation theorem when no interior point is available.

Chapter II. Normed. Linear Spaces

§ 1. Elementary Definitions and Properties Each of the function spaces mentioned in the introduction of the preceding chapter has (with one exception, A(D») a norm II I [Definition I, 3, 1] which defines the topology of major interest in the space; a neighborhood basis of a point x is the family of sets {y: Ilx- yll ~6} where 6>0. Hereafter we shall use N for a normed space, that is, a linear space in which a norm is already assigned. If the normed space is complete 1 under the metric Ilx- ylt then the space will be called a Banach space, and will generally be denoted by B. U will generally stand for the unit ball, {x: I x I ~ 1}, unless otherwise noted; the unit sphere is the set {x: Ilxll=1}. The properties we have discussed in linear topological spaces sometimes have simpler character in normed spaces. (1) A set E in a normed space is bounded if and only if it lies in some ball. (2) Call a linear operator T from one normed space N into another N' bounded if it is bounded on the unit ball U in N; define I TIl , the bound or norm of T, to be sup{IITxll: Ilxll~1}. Then: (a) IITxl1 ~IITIHxll if XEN. (b) IITII=sup{IITxll: Ilxll=1}=sup{IITxll: Ilxll 0 there is 0 such that for all T in E and all x,y in N II Tx- TYII 1 and lip + llq = 1, then Lq(fl) is linearly isometric to ll(fl)* . (5)5 If fl is a measure and if M(fl) is the set of bounded real functions

measurable on every part of S which is a countable union of sets of finite fl-measure, then M(fl) is isometric to the space V (fl)*. Note that these proofs when completed imply that all the spaces described in this section except V (fl), Co (S), and C(S), are isometric to conjugate spaces; the last two examples are closed subspaces of m(S), hence they are also complete. (6) If T is the mapping of Theorem 1 of co(S)* onto [l(S) and if V is the similar isometry of m(S) onto [1(S)*, then the mapping T* V is an isometry of m(S) onto co(S)** which coincides on co(S)~m(S) with the natural mapping Q of co(S) into co(S)**. (7) If xEm(S), then there exist x" = P"x in co(S) such that L x(s) y(s) = lim L x,,(s) y(s) for every y in [1 (S). (This is related to a SES

(j

SEa

special case of Theorem 4,3.) 4 See Banach, p. 65, for one proof for Lebesgue measure on an interval. Or approximate by functions constant on the sets of a linite family of disjoint measurable sets of linite measure, use (3), and work back. S See, for example, Hewitt and Ross, Chapter III, for the non-o--linite case.

35

§ 2. Examples of Normed Spaces

(8) For each F in m(S)* define a function T F =

1, with X 1 shows that in a Banach space B the condition (F) is equivalent to: (E') for every bounded sequence (T;,iEw) of linear operators on B, I Tix i is convergent to an element of B. In a iew

normed space N,(Fc) is equivalent to a corresponding condition (E~). (4) (a) If the w-topology is used in an LCS L, then the absolute and the unordered convergent [Cauchy] series are the same. (b) If I Xi is a series in a Banach space B, then K = sup only if

K'=sup{t~aixill: lail~1

K~K'~2K),

{II i~ Xi I :0" E L} < 00 iEif and

for all i and o"EL}0 and disjoint finite sets 'Cj (as in the proof after (2») such that setting yj= L Xi yields IIYjl1 >8 for all j in w. But

L Yj

ietj

is also weakly subseries convergent, so has a subseries with terms

jew

norm-convergent to 0; this contradiction proves that L Xi is unordered JEW convergent and subseries convergent. The main purpose of this section is to show that (2b) is not an accidental property of co(w) but is valid in all infinite-dimensional Banach spaces. Similar examples were known in all the familiar spaces, but the normed case was finally settled by Dvoretsky and Rogers in the following theorem using, for example, Cn= 1/n. Theorem 2. Let B be an irif"inite-di1flensional Banach space and let (cn>new) be any sequence of positive numbers such that L c~ < ex)" nEW

then there exists in B an unordered convergent sequence (xn>new) such that Ilxnll =Cn for every n in w.

First we prove a geometrical lemma about n-dimensional Euclidean spaces and symmetric convex bodies there. Lemma 1. Let B be an n-dimensional normed space.. then there exist points Xl' ... ' Xn of norm one in B such that for each i;;i; n and all real t 1 , ••• , tj (a)

81

§ 1. Series and Unconditional Convergence

Proof. Inscribe in C, the unit ball of B, the ellipsoid E of maximum volume. [More precisely, if Ul , ... , Un is any vector basis in B, an ellipsoid E' is the set where some positive-definite quadratic form in the components, Q(tl, ... ,tn)= L aijtitj' is ;;:;1; then the volume of E' is the i,j~n

square root of the reciprocal of determinant laijli.j~n' and E' ~ C means I i~/i Ui II ;;:; 1 if Q(t l, ... , tn);;:; 1.J By a linear transformation of B we may turn E into the Euclidean ball whose coordinates satisfy

L tf;;:; 1.

i5.n

Using subscripts to distinguish the norm with unit ball E from that with unit ball C, we want now to search for Xi with II Xi II E= II Xi I c = 1 and the Xi approximately orthogonal. By induction on i we shall find an orthonormal basis U i , i;;:; n, in the Euclidean space determined by E and points Xi' i;;:;.n, with IlxillE = Ilxillc = 1, such that (i) Xi= L aijuj , and all aij~O, and j~i

(ii) afl + ... +ail-l;;:;(i-1)/n. To begin the proof take U l =X l to be any point of contact of the surfaces of C and E, and, for the moment let U2, ... , Un' be any vectors completing an orthonormal basis in E. Suppose that Xl"",X i and Ul' ... 'U i , 1;;:;i0 the "spheroid" Ee of points whose coordinates in this basis satisfy (b)

(1 +6)"-i(exi+ ... +exr)+(1 +6+62)-i(exr+l + ... + ex;);;:; 1.

The volume of Ee is easily calculated to be greater than that of E, so there is a point in Ee outside C, and, by coming toward 0 along a ray, there is in Eea point Pe ofC-norm 1. Then Pe is not inside E, so, ifex l , ... , ex n are the coordinates of Pe' ex; ~ 1. Subtracting this from (b) gives

L

j~n

(c) [(1 +6)n-i-1J [exi + ... +exfJ + [(1 + 6+6 2)-i -1J (exf+ 1 + ... + ex;);;:;O. By compactness, there is a subsequence of 6'S tending to 0 such that the corresponding Pe converge to some point Xi+ 1 common to the surfaces of C and E. Dividing (c) by 6 and taking the limit such a sequence gives, if Xi +1 has coordinates a l' ... , an' (d)

(n-i)(ai + ... +ar)+( -i)(ar+l + ... +a;);;:;O.

Choosing Ui + l orthogonal to ul"",U i in the space spanned by these and Xi +1 and completing this sequence to a new orthonormal basis gives a representation for Xi+1 which can now be seen to satisfy the conditions (i) and (ii) for i + 1. This induction process defines Xi and U i for all i;;:; n.

82

Chapter IV. Unconditional Convergence and Bases

(a) follows from the other conditions. (ii) implies that Ilxi-udli = ~1-aii)2+ L:at)~2(i-1)/n. Since E is inside C, Ilxllc~llxllE for JO there exists me such that n>m>me imply Ilxm-xnll' ke; therefore kke'

The completeness of B implies that

I

aib i converges to some element x

iero

of B; then by uniqueness of the expansion of x, ai ={3i(x) and Ukjx= I aib i· But then Ilxm-xll' =sup II Ukx m - Ukxll ~e when m>me; kO such that klixil ~ IlxlI'~ Ilxll. This

89

§ 3. Schauder Bases in Separable Spaces

means that to prove (v) it suffices to prove that IIUrnll';£1; for this the easily verified rules of calculation stated in (iv) are required. Then I Urnll' =sup{ II Urnxll': Ilxll';£ 1} =sup{11 Uk U rnxll: Ilxll';£ 1 and kEW} =sup {IIUk xll:llxll';£1 and k;£m}. But Ilxll';£1 means IIU k xll;£1 for all k in w; hence II Urnll';£ 1. (ii) follows from this and (1c). For each x, IPn(x)lllbnll' = IIPn(x)bnll' = I Unx- Un- 1 xll';£21Ixll' so IIPnll' Ilb nll';£2. This proves that each Pi is in B*=B'*. Uniqueness of the expansion of b j proves that Pi (b) = i5ij, so (iii) holds. (vi) now follows from direct calculation: p(b)=limp(Urnb)=lim(U!p)(b) for every bin B. mEW

mew

Schauder [1J assumed that Ilbill =1 and that each PiEB*; this proof that the Pi are all continuous is due to Banach [po 111 J. Clearly a change from bi to b;=bJllbill is allowable if at the same time Pi is replaced by P;= Ilbill Pi' so the basis elements may be normalized without losing any of the properties of Theorem 1. (2) In co(w) and in IP(w), pi;; 1, the sequence (b i, iEW), for which bi =(i5ij,jEW) for each i, is a basis. (3) In C [0, 1J Schauder constructed a basis of polygonal functions: bo(t)=t; b1 (t)=1-t; b2 (t)=0 if t;£O, =2b o(t) if 0;£t;£1/2, =2b 1 (t) if 1/2;£ t;£ 1, =0 ifti;; 1; for i= 1 or 2, b2+i(t) =b 2 (2 t - i + 1); for i = 1,2,3,4, b4 +;(t)=b2(4t-i+1); ... ; for i=1, ... ,2n,b2"+;(t)=b 2(2 nt-i+1); ... Then the P's are described for each x in C [0, 1J by Po(x)=x(1); Pl(X)=X(O); P2(x)=x(1/2)-(x(0)+x(1))/2; and so on by induction. (4) Call sequences (bJ in Band (P;) in B* biorthogonal if Pi(b)=i5 ij . Then (b i) is a basis for B if and only if (i) there is a biorthogonal sequence (Pi) in B*, and (ii) (bi,iEw) spans B, and (iii) there is a K such that IIJrnPi(x) bill ;£K if Ilxll;£1 and mEW. [Banach, p. 107, Th. 3.J

Theorem 2. Let (b i) be a sequence of elements of B such that for each x there is a unique sequence a i = Pi (x) of real numbers such that (i) aib i converges weakly to x and (ii) each PiEB* or (ii') B is w-sequentially

L

iEW

complete. Then (b i) is a basis for Band (PJ is a basis for the closed linear subspace r of B* spanned by the Pi' (That is to say, a weak basis in a Banach space is a norm basis.)

Proof. Defining Urn as before, we have mEW lim P(Urnx)=p(x) for all P in B* and x in B; (ii) implies continuity of each Urn; the category theorems, [II, § 3, (1d)J assert that I Urnll is uniformly bounded. Then (4) could be applied, but a direct proof follows from Lemma II, 3, 1 which asserts that the set E of x for which the Urn X converge in the norm topology is closed and linear. But I Urnbj-b)1 =0 when mi;;j, so E contains all bj ; since E is weakly closed, it contains all of B. The same argument applies to the U!Pi'

90

Chapter IV. Unconditional Convergence and Bases

If instead of (ii) the condition (ii') holds, follow the proof of Theorem 1 down to the completeness of B' to get continuity of each [3i' that is, (ii). (w-convergence and w-completeness of B are used in the third paragraph, where norm completeness of B sufficed under the stronger hypotheses of Theorem 1.) Singer, p. 209, reports that Bessaga and Pelczynski have proved Banach's statement that (i) suffices for this conclusion. Definition 3. A basis (b;) of B is called boundedly complete in B if for each sequence (a i) of real numbers such that the sequence (L~n ai bi II' nEW) is bounded in B there is an x in B such that ai = [3Jx) for all i, so X= aib i. A basis (b;) is called a shrinking basis for B if for each [3 in

L

iew

B*,limpn([3)=O where Pm([3)=norm of [3 restricted to the range of nEW Vm; that is, Pm ([3) = sup {[3(X): x = Vmx and Ilxll ~ 1}. Next we investigate when ([3;) is a basis, not just a w*-basis, for all of B*. Lemma 1. If (b i) is a basis for B, then the following conditions are equivalent: (i) ([3i) is a basis for B*. (ii) (v,:, mEW) tends to zero in the strong operator topology. (iii) (b;) is a shrinking basis for B. (iv) ([3;) spans B*. Proof. From Theorem 1, (vi), ([3(b;), iEW) is the only possible coefficient sequence for [3; hence ([3;) is a basis for B* if and only if I [311--+0 for all [3, so (i) and (ii) are equivalent. (iv) and (i) are equivalent by the last part of Theorem 2.

V,:

Clearly Pm ([3) = sup{[3(x): x = Vmx and Ilxll ~ 1} ~sup{[3(Vmx): Ilxll ~1} =sup{ Vm* [3(x): Ilxll ~ 1} = IlVm* [311· But {Vmx: Ilxll~1}S{x:x=Vmx and Ilxll~K} where K is a common bound for IlVmll, so IIV': [311 ~ K Pm ([3). Hence (ii) and (iii) are equivalent. Lemma 2. Let (b;) be a monotone basis for B, let ([3i) be the corresponding biorthogonal sequence in B*, and let r be the closed linear space spanned by the [3i' Then (i), (ii), and (iii) below are equivalent and imply (iv): (i) (b i) is a boundedly complete basis. (ii) For each F in B** the series L F([3i)b i converges to a point YF of B with IIYFII ~ IIFII. iew

§ 3. Schauder Bases in Separable Spaces

91

(iii) Q B n r.L = {o} and the projection T of B** onto Q B along r.L is of norm 1. (iv) B is isometric to a conjugate space (= r*). 1 Proof. If (i) holds, take F in B** with IIFII ~ 1. Then by Theorem II, 5,4 there exists a sequence (Yk' kEw) such that IIYkl1 ~ 1 for each k and lim/MYk)=F(Pi) for all i in w; hence limY(Yk)= F(y) for every y in r. kEW

kEW

Then for each n, IF(Pi)bi=lim IPi(Yk)bi=limUnYk' But IIUnll~1, i;S;n

so Ltf(P;)bill

kEW

i::;n

kEW

=1i~IIUnYkll~lirpE~~PIIUnIIIIYkll~1.

By (i) there exists

YF in B satisfying (ii). If (ii) holds, let TF=QYF; then T is linear and II TFII = IIYFII ~ IIFII if FEB**. Also F-TF vanishes on each Pi; therefore F-TFEr.L. If GEr.L, then TG=O, so T2=T; also, if TF=O, all F(PJ=O or FEr.L. Hence T is a projection of norm 1 of B** on QB along r.L. If (iii) holds, let (aJ be a sequence of real numbers such that if Xn = I aib i , then (Xn' nEw) is bounded, say Ilxnll ~ 1 for all n in w. Then lim·Pi(xn)=a i nEW

for all i so [Th. II, 3,2] for every yin r, limy(xn)=cp(y) exists; cP is in nEW

r*, and Ilcpll ~ 1. Take F in B** so that IIFII = Ilcpll and F is an extension of cp; let y=Q-l TF. Then F - TFEr.L, so y(y)= TF(y)=cp(y)=limy(xn); nEW

in particular, Pi(Y) = lim PJx n) = ai' Hence the expansion of Y in the nEW

basis (b i ) is

I

aib i and (iii) implies (i).

iEW

B is isometric to QB which by (iii) is isometric to B** / r.L. But by Lemma 11,1,2, B**/r.L is isometric to r*, so (iii) implies (iv). Turn now to the connections between bases and reflexivity; much of the rest of this work comes directly from R. C. James [1,2]. Theorem 3. If (bJ is a basis for B, then B is reflexive if and only

if the basis is both shrinking and boundedly complete. Proof. Suppose that B is reflexive. Then the w- and w*-topologies agree in B**; by Theorem 1, (vi) and Theorem 2, (PJ is a basis for B* and by Lemma 1, (bJ is shrinking. Since r.L={O}, Theorem 1, (ii), and Lemma 2 imply that (bJ is boundedly complete. If on the other hand, (bJ is shrinking and boundedly complete, by Lemma 1, r=B*, so r.L={O}. By Lemma 2, QB=B**. 1

Alaoglu observed that (i) implies (iv).

92

Chapter IV. Unconditional Convergence and Bases

Corollary 1. Let (bJ be a shrinking basis for B and let (f3i) be the corresponding basis for B*; then (i) (f3i) is a monotone basis if and only if (bJ is monotone basis, and (ii) (f3J is a boundedly complete basis. Proof. (i) follows from Lemma 1 and (1). (ii) can be proved as "(iii) implies (i)" of Lemma 2 was proved, or as follows: Let B' = B*; then (f3i) is a basis and (QbJ is the corresponding biorthogonal system in B'*=B**, so r'=QB. Q'Q* is the projection of B***=B'** onto Q' B' along r'J.. and is of norm 1. By "(iii) implies (i)" of Lemma 2, (f3J is a boundedly complete basis for B* = B'. The next theorem describes B** when (bJ is a shrinking basis. Theorem 4. Let (bJ be a monotone shrinking basis for B and let (f3i) be the corresponding boundedly complete basis for B*; then: (i) Each sequence of numbers (d i) such that

~~E t~n di bi II < 00

deter-

mines an element F of B** by means of the relation F(f3)-= L dic i whenever 13= L ci f3i. JEW iew

(ii) If FEB** and di=F(f3i) for every i in w, then IIFII

and F(f3) =

L djc i whenever 13= L ci f3j· iew

=~iE~ II i~ndibi II -

iew

Proof. Most of this conclusion is in the assertion that (Qb j, iEW) is a w*-basis for B**. This follows from Theorem 1, (vi), lifted one space, for Lemma 1 assures us that (f3J is a basis for B* and that its sequence of coefficient functionals is (Q bJ We give next an example of R. C. James [2J which displays a nonreflexive space isometric to its second conjugate space although not under the natural isometry; the deficiency of QB in B** is precisely 1. Example. If E is the set of finite subsets (l of the integers, define functions T from R W into 12(w) as follows: If x=(x(i), iEW), and (l consists of the integers ii < i2 < ... < in' then

I;,.xU)

=x(ij+1)-x(i)if1~j00, (i) and (ii) imply that Ilxl12~2lx(kW for each x in Band kin w.

§ 3. Schauder Bases in Separable Spaces

93

(c) Under this norm B is a Banach space. (d) If Ei is the subspace of B containing just those elements x for which x(k)=O if k is congruent to i mod 2, then each Ei is isomorphic to 12 (W), and is closed in B; EI n E2 = {O}, but, James [2] to the contrary, E 1+ E2 is not all of B. James [2] shows that the unit vectors in B form a monotone, shrinking, but not boundedly complete basis in B. From Theorem 4 he shows that B** is isometric to the space of sequences satisfying condition (i), and that Q(B) is embedded in it in the original way as the space of sequences satisfying both conditions (i) and (ii); hence the deficiency of Q(B) in B** is precisely one. The isometry between B** and B is set up as follows: If X in B** corresponds to a sequence (d i , iEW) satisfying (i), let d be the limit of (dJ and let TX=(Xi) where XI = -d, and Xi+1 =di-d for i=1,2, ... The literature on bases has grown enormously since 1958. Fortunately there exists a comprehensive account of bases and related systems by Singer; for those interested in applications of bases there is the excellent survey by V. D. Mil'man [10] in Russian or in English translation. Various results are known about basic sequences, that is, sequences which are bases for the subspaces they span. For example, (Day [15]) from a point-to-set version of Borsuk's antipodal-point theorem one can prove that for each finite-dimensional subspace E of N and each subspace F of larger dimension than E that there is X in F such that Ilx+EII = Ilxll. (See Krein-Krasnoselski-Mil'man whose result is more accessible in Gohberg-Krein, Th. 1.1 and Cor.) From this it is possible to get Day [15]' Theorem 5. If N is an infinite dimensional normed space and if (c m ) is a sequence of positive numbers, then there exists biorthogonal sequences (b i) in Nand (P;) in N* such that (i) Ilbll = 1 = IIPil1 for all i in w, (ii) (b;) is a basis for the closed linear manifold L which it spans in B, and (iii) if Um(x) = Pi(x)b i, then Urn is a projection in L of norm S 1 + cm. [Note

L

i~m

that, from the proof in Day [15], if N is so large that there is no countable total subset in N*, then the Cm can all be taken to be 0.] V. D. Mil'man [10] calls a basis in which Ilbill = 1 = IIPil1 for all i in wan Auerbach basis and a basis in which IIUm ll-1 an asymptotically monotone basis. He gives many conditions on a sequence (Xi) sufficient to make it possible to select a basic sequence from (x;), and illustrates how many topological proper:ties of normed spaces follow from these ways to choose bases for subspaces. The Approximation problem: Enflo's counterexample. We close this section with a report on the recent negative answer to Schauder's question:

Chapter IV. Unconditional Convergence and Bases

94

Does every separable Banach space have a basis [Banach, p. 111 J. This requires some description of work of Grothendieck [3, Chap. 1, § 5.] The example of Enflo is too complicated to give here but is to be found in Enflo [2J. In his analysis of tensor products Grothendieck isolated an important property which all the familiar Banach spaces have. The several variants are successively stronger restrictions on B; for separable spaces the bounded approximation property is formally weaker than the presence of a Schauder basis.

Definition 4. An LCS L, in particular, a normed space, is said to have the approximation property (AP) if there is a net (Tn' n E,1) of continuous linear operators of finite rank in L such that (Tn) converges to the identity operator uniformly on each totally bounded set K in L. A normed space N has the bounded approximation property (BAP) if there is such a net with a uniform bound on I T"II; N has the metric approximation property (MAP) if there is such a net with II T"II :::;; 1 for all n in ,1. (6) Grothendieck [3, page 167] lists a number of properties of a space B equivalent to some of these, mostly involving tensor products of B with other spaces, and he uses these to prove: (a) If B* has (AP), so does B. (b) B* has (AP) if and only if for each Banach space F, each compact linear map T of B into F is the limit in the norm topology of L(B,F) of elements of B* &iF; that is, of operators of finite rank from B into F. (c) On p. 180 Grothendieck shows simply that every LP(Jl), 1 :::;;p:::;; 00, has (MAP). Then by (a) every abstract L-space and abstract M-space (VI, § 4) has (MAP). (c) Clearly every Banach space with a basis has (BAP). (7) Grothendieck [3, page 170] then gives a number of conjectures equivalent to the conjecture: Every Banach space has the approximation property. Some of these (p. 171) are quite special properties of particular spaces, such as his (f) for every compact measure space (X, Jl) and every continuous function K on X x X such that S K(x,s)K(s,y)dJl(s)=O x it follows that S K(s,s)dJl(S) =0, or (f"). If uEzt xC o and u 2 =0, then x

the trace of u is O. (8) (a) Grothendieck [3, p. 181] shows that if a reflexive space has (AP), then Band B* have (MAP). (b) Rosenthal observed that if B* is a separable conjugate space with (AP), then Band B* have (BAP). (9) Enflo [2] has constructed a subspace of Co which does not have (BAP). By (8b) the dual B*, which is a factor space of 11 and is therefore separable, does not have (AP). These spaces Band B* are counter examples to show also that separable spaces need not have a basis.

§ 4. Unconditional Bases

95

(10) Pelczynski [3J has proved that a separable B has the bounded approximation property if and only if it is a complemented subspace of a space with a Schauder basis.

§ 4. Unconditional Bases 2 Definition 1. A basis (b i, iEW) in B is called an unconditional (absolute) Pi(x)b i is unconditionally basis for B if and only iffor each x the series (absolutely) convergent to x. iEW The first theorem of this section is analogous to that of the preceding section. If L is the set of finite subsets v of w, define Uv x = Pi (x) b i

I

I

iev

and U91 x=o for all x. Then define Vv=i- Uv and W v=i-2Uv = V v - U v' Theorem 1. Let (b i, iEW) be an unconditional basis for B; let Ilxll'=sup{IIUvxll:vEL}; let Ilxll"=sup{IIu\.xll:vEL}. Let B' and B" be the spaces obtained by renorming B with 11 ... 11' and 11 ... 11". Then (i) B' and B" are Banach spaces isomorphic to B. (ii) Every rearrangement of (b i) is a monotone basis for B' and for B". (iii) Each Pi is the unique element of B* for which Pi(bj)=Dij for all j in w, and IIPill' Ilbill' = 1 for all i in w. (iv) If j1 and VEL, then UI'UV =Ul'nV' so for each j1, U~=UI' and WI'2 = i. Setting A= symmetric difference 3 of j1 and v, WI' W. = W;.. (This implies that every U I' and every VI' is a projection and that every WI' is an involution 4 in B.) (v) U I' x tends to x and VI' x tends to zero for each x in B. Also IIUI'II'~1 and IIWI'II"~1, so IIUI'II"~1 and 11V1'11"~1for all j1 in L. If j1 and V are disjoint and non-empty, if x = U I'x =1=0, and if y = Uv Y =1=0, then II x + t y II" is an even, convex 5 function of t, so it is a non-decreasing function of Itl. (vi) (Pi' iEW) is a w*-unconditional basis for B*; that is, for each P in B* the series I P(b i) Pi is w*-unordered (even w*-absolute) convergent to p. iEW The difficult part of this proof follows the pattern of Theorem 1 of the preceding section. The rest follows by exploiting the involutions WI" j1EL. 2 This terminology is that of James [1,2], not that of Karlin [1] or Gelbaum. By § 1, (1 d) any of the definitions (B)-(D) can be used to define unconditional convergence. We shall generally use unordered convergence. 3 The symmetric difference of two sets is the union minus the intersection. 4 An involution is a linear operator whose square is the identity. 5 If a Banach space has a basis with this property, then the expansion of each element is unordered Cauchy, so the basis is unconditional.

96

Chapter IV. Unconditional Convergence and Bases

For unconditional bases, there are alternative conditions equivalent to bounded completeness or shrinking. The usual bases in co(w) and in IP(w), p~i, are unconditional; that defined by Schauder [i] in C[O,i] is conditional (see Corollary i) and the basis in James' example at the end of the preceding section achieves its purpose because it is conditional. Theorem 2. If (bJ is an unconditional basis for B, then the following conditions are equivalent: (a) The basis is boundedly complete. (b) The space is weakly sequentially complete. (c) There is no subspace of B isomorphic to co(w). Proof. Without loss of generality, we may assume that the basis and the norm in B have the relationships expressed in Theorem i, (v), for II ... II"; this might be called unconditional monotony of the basis. To prove (c)=>(a) suppose that there is a sequence (a i) such that C~n aib i, nEW) is a bounded sequence in B which does not converge to any element of B. Then it can be assumed that II

L ai bi II < i

for every

IEIL

fJ in l:. Since the series is not Cauchy in B, there exist sequences of integers (n k) and (md and a positive number d such that nk < mk < nk+ 1 for every k in wand, setting zk=sum of all aib i with nk;£i J1(e) = 1 = 11J111,

while Ile-x/llxll I ~ 1.] Also, if x is in the subspace co(w) of m(w), then I Tn xll-+O, so J1(T" x)-+O. By invariance J1(X)=O if xeco(w). Now if yem(w), there exists x in co(w) such that lim sup Yn ~ sup (xn + Yn) ~ inf (xn + Yn) ~ lim inf Yn' nEW nEW nEW nEW

§ 2. Fixed-point Theorems

109

Then ,u(y) = ,u(x + y) and

1 _winf Yn. sup Yn\~,u(x+ y)=,u(y)~lim lim_w sup Yn=,u(elim_w ~

Theorem 3 [Markov]. Let K be a compact, convex subset of an LCS L and let (Fs' SES) be a commuting family of continuous functions from K into itself such that each Fs is affine; that is, for x, Y in K and 0 < t < 1, Fs(tx+(1-t)y)=tF s(x)+(1-t)F s(Y). Then the Fs have a common fixed point in K. Proof. Let Ks be the set of fixed points of Fs. Theorem 1 asserts that each Ks is non-empty; affineness implies that Ks is convex, continuity that Ks is closed; hence each Ks shares the properties of K including compactness. But commutativity of Fs and Fo s, t in S, imply that Fs(KI)~KI' By Theorem 1, Fs has a fixed point in K 1, so KsnKI again shares the properties assumed for K. By induction on the number of elements, Ks is non-empty for each finite subset a of S. By compactness,

n

nKs

SEU

K' =

=1=

0; any element of K' is a fixed point common to all the Fs.

SES

There is a simple direct proof of Theorem 3, probably due to Bourbaki, but certainly inspired by F. Riesz's proof of the ergodic theorem [1]. For a in L, the set of finite subsets of S, and n in w define

Fun =

n I(~n F~)l. l n J

SEU

By compactness of K, for each x in K the net (Fun x, (a,n)EL x w) has a convergent subnet (Fumnmx, mELJ) with some limit y. Then y is a fixed point of every F s ' for, if sEam and rm =a m\{s},

F - -I' F .[F:m+lx_ Fsx]_ sY Y - mELl 1m r n - 0, m m nm because the expression after the limit is in (K - K)/n m ; by I, 4, (17b), this tends to 0 as nm -> (1). Banach p. 33 proved that the additive group R is amenable by careful construction of a sublinear p to use in the Hahn-Banach theorem. Day [12] first adapted that proof for the following result. Theorem 4. Every abelian semigroup is amenable (left and right). Proof. Apply Th. 3 to the set M of means on L (as in the proof of Th. 2) and the family {1/:aEL} restricted to M. Definition 3. If K is a convex compact subset of an LCS L, let d(K) be the semigroup of affine continuous transformations of K into K, with composition for multiplication.

Chapter V. Compact Convex Sets and Continuous Function Spaces

110

Theorem 5 [Day 17]. A semigroup L: is left amenable if and only K of each LCS L and for each homomorphism h of L: into d(K) there is in K a common fixed point p of all

if for each compact convex subset the transformations h(1'

kemark. There is also a version of this for topological semigroups; see Day [18], § 6. Proof. Sufficiency is simple because the choice K = set M of means on m(L:), h(1= I:IK describes a homomorphism of L: into d(M) whose fixed points are the left-invariant means. For the converse, choose y arbitrarily in K and define T from L*IK ~A(K) by [TqJ](O")=qJ(h(1Y) for all 0" in L:. Then T* carries means on m(L:) to means on A(K), and, in particular, nT*6,=h,y for each r in L:. If J1. is a left-invariant mean on m(L:), then V= T* J1. is a mean on A(K) and setting p=n(v) gives a point of K such that v(qJ) = qJ(p) for all qJ in A(K). By (1), there are finite averages qJn= Ia m 6, such that J1.=w*-limqJn. We can calculate that 1:6,=6(1' so neA

r

Then w*-lim T*('" ~ a nr 6at)= T* 1* a J1. = T* J1. = w*-lim T*'" i..J ant 6 ,p n

n

or for each qJ in L*,

li~ qJ(Iam h(1'Y)= qJ(n T* I: J1.) = qJ(n T* J1.) = qJ(p) but the first quantity also equals

so p is invariant under all h(1' because L* separates points of K. Theorems 2 and 4 imposed some structure conditions on the semigroups of affine continuous mappings. Other theorems impose some more geometric conditions on K or L:. The next important fixed point theorem is due to Ryll-Nardzewski; the present proof is from Asplund and Namioka. Definition 4. A semigroup Y' of transformations of a subset E of an LCS L is called distal (sometimes noncontracting) if whenever x =l= y, then 0 is not in the closure of {S x - S y: S EY'}. (3) In an LCS L a semigroup Y' is distal if and only if there is a prenorm p and a number d>O such that p(Sx-Sy»d for all S in Y'.

§ 2. Fixed-point Theorems

111

Theorem 6 (Ryll-Nardzewski). Let K be a convex, w-compact subset of an LCS L. Then each distal semigroup !f of w-continuous affine maps of K into itself has a common fixed point p in K. Proof. It suffices to prove that each finite subset T1 , ••• , T" of !f has a common fixed point, so we begin with the case where the T; generate !f, and let T=(TI + ... + T,,)jn. T has a fixed point a. If all T;a=a, we are finished; if not, discard those i for which the equality holds, renumber, and assume we are now in the case where T; a 01= a for all is n. As !f is distal, there exist a prenorm p and d > 0 such that (i) p(S T;a-Sa»d for all S in !f and iSn. Let W be the closed convex hull of the orbit {Sa:SE!f}. By 1, (9a), there is a convex C~ W such that the p-diameter of W\C r, is the intersection of closed balls with K so is convex and w-compact; hence Er= nEt t>r

112

Chapter V. Compact Convex Sets and Continuous Function Spaces

is non-empty.] (b) A set A ~ N is said to have normal structure if for each closed convex bounded set W in A with more than one point there is a point p in W such that rp(W) 0. (b) If S is locally compact, Hausdorff, and if Co(S) is the space of functions in qS) which vanish at infinity, then Co(S) is also an (A B)-lattice but its positive cone has no core point. co(S) is a simple example. (c) All the spaces IP(S), with p ~ 1, and with K containing the non-negative elements of IP(S), are (AB)-lattices. (9) (a) In a normed lattice V, (A) is equivalent to (A') for all x in V, Ilx+ + x-II = Ilxll, and (B) is equivalent to (B/) if x and YEK, then Ilxvyll~llxllvllyll. (b) (B) implies that order intervals are norm-!:>ounded, so V* ~ V'. (c) If the cone K in a normed lattice satisfying (A) is complete under convergence of monotone Cauchy sequences, then V' ~ V*. [Kaplansky, conversation about 1949. If it could happen that for fin K+, f(U) is unbounded, then f( Un K) is unbounded by (a) so there exist Xn in UnK such that for all n,f(xn»4n. Then x=Ixn/2n is in KnU n

and f(x) ~f(xJ2n) > 2n for all n!J (d) Continuity of v implies that the positive cone in a normed lattice is closed. Lemma 1. An (A B)-lattice is a normed lattice.

By (2 d), for x, y in V, x v y = (x - y) + + y; the vector operations in V are continuous in both their variables, so continuity of v is equi-

§ 1. Vector Lattices and Normed Lattices

129

valent to continuity of ( t. But by sublinearity v=x+ -y+ ~(x-yt and -v=y+ -x+ ~(y-xt =(x-y)-. Hence Ilx+ - y+ II = Ilvll = Ilv+ +v-II ~ II(x- y)+ +(x- y)-II = Ilx- YII· (10) Let V be an (AB)-lattice with positive cone K and unit ball U; then: (a) If fEK+, then Ilfll = sup {f(x): XEK n U}. (b) Hence if fEV*, Ilf+11 ~ Ilfll and f+EV*. (c) Hence V* is a vector lattice and (d) V* satisfies condition (B). Lemma 2. If V is an (AB)-lattice, then V* is an (AB)-lattice in which every set with an upper bound has a least upper bound. Proof. (10) asserts that V* is a Banach lattice with property (B). To prove that V* has property (A), use (10a) and (6) to get 1If+ + f-II = sup{f+(x)+ f-(x): XE U} = sup{f+(x)+ f-(x): XEU nK} = sup{sup{f(y): O~y~x} +sup{ - f(z): O~z~x}: XEK n U}. Set W=YI\Z, y'=y-w, z'=z-w, x'=y'-z'; then the quantity in the outer braces is sup {f(y): O~y~x} +sup{ - f(z): O~z~x} = sup {f(y)- f(z): O~y,z~x} = sup {f(y')- f(z'): 0= y' I\z'~y' V Z' ~x}. But by (A) and (B) we have Ilx'll = 11y' +z'll = 11y' v z'll ~ Ilxll, so the sup above becomes sup{f(x'): x H +x'- ~x}.

Now taking the sup on x in K n U Ilf+ + f-II = sup{f(x'): x'+ +x'-EKn U} = sup {f(x'): x' E U} = Ilfll. This says that V* satisfies (A') which is equivalent to (A) by (9 a). If A is a subset of V* which has an upper bound bo, let B = {b: A ~ b ~bo}, and if bEB let Bb=Bn(b-K+). Then intervals a~f~bo are norm-bounded, therefore compact in the w*-topology, so B =

n

aEA

(a+K+)n(bo-K+) is w*-compact. Because V* is a lattice, the Bb have non-empty finite intersections and are themselves w*-c1osed; hence Bb contains some point b1 • Then A ~ b1 ~ B, but if b 2 E V* and

n

bEB

130

Chapter VI. Norm and Order

A~b2'

then b o /\b 2 EB so

bl~b2;

that is, bi is the least upper bound

of A. Theorem 1. If V is an (A B)-lattice, then V* and V** are boundedly complete (A B)-lattices. The natural mapping Q of V into V** is not only a linear isometry into V** but is also a lattice isomorphism; that is, Q(x v y) = Qx v Qy. Proof. Q preserves order, so Q(x v y) ~ Qx v Qy. To prove equality for all x, y it suffices to prove it for the case where x /\ y = o. By (6f) raised one space we have, for each f~O in V*, (Qx v Qy)(f) = sup{g(x)+h(y): g,h~O and g +h= f}.

But (3) asserts that there exist

g,h~O

with g+h=f and g(x)=f(x),

h(y) = f(y), so this gives (QxvQy)(f)~f(x)+ f(y)=f(x+y)=f(xv

y)= [Q(xv y)](f).

This proves that Qx v Qy = Q(x v y). Corollary 1. Every (A B)-lattice has a norm-completion which is also an (AB)-lattice. Definition 6. (a) A Banach lattice is called an (AM)-space (abstract m-space) if the norm and order in the space are related by the conditions (A) and (M) If x,y~O then Ilx v yll = Ilxll v Ilyli. (b) A Banach lattice is an (AL)-space (abstract Lebesgue space) if order and norm are related by (A) and (L) If x,y~O then Ilx+yll=llxll+llyll. Clearly either (M) or (L) implies (B), so every (AM)-space, and every (AL)-space, is an (AB)-lattice. Theorem 2 [Kakutani 2]. The conjugate space of an (AM)-space is an (AL)-space. The conjugate space of an (AL)-space, and, therefore, the second conjugate space of an (AM)-space, is a boundedly complete vector lattice which is an (AM)-space whose positive cone has an interior point u such that the unit ball is the order interval {x: - u ~ x ~ u}.

[More can be added to this with the proof of Kakutani's Representation Theorem 2,2.] Proof. In the presence of (A) and (B) (which follows from either (M) or (L)), it suffices to check (L) or (M) in V* for positive elements. Then all that is not immediately verifiable here is the existence of uin V* if V is an (AL)-space. But the positive cone C in an (AL)-space is a maximal cone in which the norm is additive; hence there is u in V* such

§ 2. Linear Sublattices of Continuous Function Spaces

131

that C= W(u). Since for fin U", 1If+ II v Ilf-il = Ilfll, and Ilxll =u(x) ~f(x)~-u(x) if XEC, u~f+~f~-f-~-u whenever Ilfll~1. Conversely, u~f~ -u implies u~f+ and u~f-, so Ilull ~ Ilf+ II v Ilf-il = Ilfll. (11) Suppose that V is a normed vector lattice. (a) If every order interval of V is contained in a ball of V, then V* ~ V'. (b) If f in V' implies that f is bounded on C n U, the positive part of the unit ball (in particular, if each ball is contained in some order interval) then V' ~ V*. (See Klee [9J for improvements on these and later references.) (12) Let V be a vector lattice with a cone C which has a core point u; let Ilxllu=inf{A:Au~x~-Au}. Then: (a) This function is a norm in V. (b) This norm in V makes Van (AM)-space if the space is complete. (c) The unit ball is (u-qn(-u+q. Hence (d) a linear functional f on V is continuous if and only if it is bounded on order intervals. (e) If v is another core point of C, then II Ilu and II Ilv are isomorphic, that is, they define the same topology in V. (13) Properties (A) and (B) are shared by all the common Banach function lattices, LP(Il), p ~ 1, C(S), Co(S). (M) and (L) are very special properties; to show this we give the next section some representation theorems of Krein and Krein t1,2J and Kakutani [2J. The books of Peressini and Schaefer [1 J give much more information on ordered linear. topological spaces and ordered normed spaces.

§ 2. Linear Sublattices of Continuous Function Spaces We give in this section Kakutani's characterization [2J of closed sublattices of continuous function spaces; they are isometric to (AM)spaces. We give first conditions identifying the closed vector sublattices of a given C(S) in terms of the linear relations connecting the values of the functions at different points.

Definition 1. (a) If S is a set, call S2 x R2 the set of (two-point) linear relations over S and denote it by A. Then A +, the set of non-negative linear relations over S, is the subset of those (s, S', r, r') in A such that rr' ~O. (b) If A is a set of real-valued functions on S define A(A), the set of linear relations satisfied by A, to be {(s, S', r, r'): r x(s) = r' X(S') for all x in A},

let A o(A)={(S,SI)xR 2 :x(s)=O=X(S') for all x in A}, and let A+(A) =A(A)nA+. (c) Dually, if A' is a subset of A and A a subset of real functions on S, then A(A') is the set of all those functions x in A such that r' X(S') = rx(s) for all (s,s',r,r') in A'.

Chapter VI. Norm and Order

132

(1) Let A be a subset of C(S), where S is a compact Hausdorff space. Then: (a) If Ass,(A) = {(r,r'): (s,s',r,r')EA(A)}, then for each (s,s') in S2 the set Ass,(A) is a linear subspace of R2. (b) If (s,s',r,r')EA(A), so is (s',s,r',r). (c) If (s,s',r,r') and (s',s",t,t')EA(A), so is (s,s",rt, r' t'), (d) If A' ~ A, then A(A')~ A(A). (e) A (linear hull of A) = A(A), (f) A (closure of A) = A(A). (g) A (smallest linear sublattice of C(S) containing A) = A + (A) U Ao(A). (2) Let C=C(S), S compact Hausdorff; then: (a) If A'~A"~A, then C(A')~C(A")~C(A)={O}. (b) If A'~A, then A(C(A'»)~A'. (c) If A' ~ A, then C(A') is a closed linear subspace of C If A' ~ A +, then C(A') is a closed linear sublattice of C Lemma 1. If A~C=C(S), S compact Hausdorff, then C(A+(A») is L o ' the smallest closed linear sub lattice of C which contains A, Proof. C(A +(A») ~ C(A(A») ~ A, and C(A +(A») is a closed linear sublattice of C, so Lo ~ C(A + (A»). To show that each x in C(A + (A») can be matched with an element of Lo we begin with a less ambitious approximation, (a) Given x in C(A + (A»), for each sand s' in S there is a yin Lo such that y(s)=x(s) and y(s')=x(s'). Let F be defined from C into R2 by F z = (z(s), z(s'»). Since F is linear, (a) is equivalent to the restriction F( C(A + (A»)) ~ F(Lo)' If F(A) contains two linearly independent elements, then F(Lo)=R2, F(A) is contained in a line A={(r,r'):pr-qr'=O} if and only if (s,s',p,q) is in A(A). If pq x(s) - 1::. As s" varies over S this set of neighborhoods covers S; by compactness 'there exist SI"'" Sn and corresponding neighborhoods VI"'" Vn and elements Yl"'" Yn of Lo such that UVj=S; if w = V yj, then w(s')=x(s') and w(s)"?,yj(s»x(s)-I:: i =:;n

i 5:n

for ails in S. Now to each s' in S there is such a w that is never far below x and agrees with x at s'; hence there is a neighborhood V' in which w(s)1. Then xi>O so new

,L Ilxill = II,L Xi I =

'~n

hence

LXi

Ilanll

l~n

~k;

is an absolutely convergent series in V which must have

i

a sum b l =lima n. If aEA, then new k = Ilblll ~ Ilblll + II(b l va) -bill = Ilb l vall =

11(~ie~an) vall = I ~ie~(an v a) I ~ k.

Hence II(b l va)-blll=O so b l va=b l if aEA. But each element of Al is below an element of A, so b l is the least upper bound for AI' and hence for Au' This property was used by Kakutani [1] to find a Boolean algebra of elements of an (AL)-space V as a step in representing V as a space of functions summable with respect to some measure. Call an element v in Va Freudenthal unit (or F-unit) for V if v /\ x=o implies x=O. Theorem 2. For each (AL)-space V there is a set S, a Boolean a-algebra !f of subsets of S, and a measure J1. on !f such that V is isometric and isomorphic (both linearly and latticially) to LI (J1.). In case V has a Freudenthal unit, S may be taken compact H ausdorJJ, J1.(S) finite (even one), and !f chosen so each of its elements differs by a set of J1.-measure zero from an open-and-closed subset of S. If V has no Freundenthal unit, S may be taken as a union (necessarily uncountable) of such compact HausdorJJ spaces Sp !f the family of countable unions of elements of the

corresponding

g;, with J1.

(U EtJ = ,L J1.t,(Et.). IECI)

lEW

The proof is to be found in Kakutani [1]. In the case where an F -unit v exists, for each set A of non-negative elements of V, Theorem 1 asserts that there is a largest a' ~ v such that a' /\ a = 0 for all a in A. The family of all such a' is a Boolean a-algebra, even a complete Boolean algebra, with a measure J1.'(a') = Ila'll. This can be represented as in Stone [1]. In the general case, the family T is a maximal family of positive elements t of V for which t /\ t' =0 whenever t t'. This splits V into an II (T)-sum of (AL)-spaces Vt ~ V, where XE VI if and only if x+ /\ t' and X- /\ t' are 0 for each t' *t. (1) Some results of Maharam imply that each (AL)-space V with an F-unit can be represented as an II(ru) sum of a sequence of spaces LI (J1.n), where J1. u is an atomic measure on all subsets of some finite or

*

§ 4. Special Properties of (AL)-Spaces

139

countable set So' and where each {In' n > 0, is a scalar multiple of product measure on a compact discontinuum Sn=2 En, where if n>m, the cardinal number of En> that of Em~ Xu. See Day [6] where this is applied to show that every (AL)-space is isomorphic to a rotund space (VII, §2).

Consequences of Theorem 2 and the well-known properties of V({I) spaces are other theorems of Kakutani [1]. Theorem 3. Every interval {x:a~x~b} in an (AL)-space is wcompact.

Theorem 4. Every (AL)-space is wow-complete. Kakutani shows in the proof of Theorem 2 that every separable (AL)-space is isometric and isomorphic with a Banach sublattice of Ll = Ll ({I), {I Lebesgue measure on [0,1]. Then Theorem 4 follows from the corresponding theorem proved for V by Steinhaus, and Theorem 3 from the corresponding theorem proved for Ll by Lebesgue. (See Banach, p. 136.) Theorem 5 (Dunford-Pettis). If Tl and T2 are wcc linear operators from an (AL)- [or (AM)-] space V into itself, then T 1 oT2 is norm cc. This was proved by Dunford and Pettis for Ll({I), {I Lebesgue measure. Generalizations and later references can be found in Bartle, Dunford and Schwartz and in Grothendieck [5]. Once it is known for every (AL)-space, Theorems III, 3,2 and VI, 1, 2 show that it is true in every (AM)-space. The proof in Ll({I) depends on the result of Dunford-Pettis that a wcc linear operator from Ll({I) into itself carries each weakly convergent sequence into a norm-convergent sequence. This fact, in turn, is largely dependent on the Orlicz-Pettis theorem IV, 1, 1. A proof of the following generalization, due to Kakutani [2], of the Riesz representation theorem for linear functionals can be found in Halmos [1], Chapter 10.

Theorem 6. Let S be a compact Hausdorff space and let !/ be the family of all Baire sets in S .. that is, !/ is the Boolean a-algebra of subsets of S generated by the compact Go subsets of S. Then each f in C(S)* determines a unique countably-additive real function {I defined on !/ such that f(x) = Jxd{l and Ilfll = total variation of {I. s

The construction is essentially that described by Riesz and Nagy, a variant of the Daniell integral. (2) There have been many generalizations of Riesz's original representation theorem for linear functionals on C [0,1]. Gel'fand and Dunford-Pettis gave many representations for linear operators between

140

Chapter VI. Norm and Order

familiar function spaces. More recently, Grothendieck [5] and Bartle, Dunford, and Schwartz (see these papers for further references) have discussed wcc linear operators from C(S) into a Banach space B. Basically, the B-D-S result is that such a T is representable as Tx = Sx dqJ, S

where qJ is a vector measure; that is, a function defined on a suitable a-algebra [I' of subsets of S into B such that for each f in B*, f OqJ is an ordinary real-valued, countably additive set-function on [1'. B-D-S also show that qJ has this property if and only if the image of the unit ball V" of B* under the mapping 1>(f) = f OqJ is a relatively w-compact subset of ca([I'), the Banach space of all countably additive, real-valued set functions tjJ on [I' with II tjJ II = total variation of tjJ. (3) (a) A corollary of Theorem 6 and of the Lebesgue dominated convergence theorem is: A sequence (xn' nEW) is w-convergent to x in C(S), where S is a compact Hausdorff space, if and only if {llxnll: nEW} is bounded and limxn(s) = x(s) for every s in S. [See Banach, p. 224, new

Th. 8 for metric S.] (b) Smul'yan [9] gives an examples of a net in C(S) which is bounded and pointwise convergent to zero but is not weakly convergent. 4 (a) Grothendieck [5] discovered another unusual property of m(S)*; (M w) Every w*-convergent sequence (5 i , iE w) is w-convergent. See III, 4, (8). (It suffices to assume that w*-lim 5 i =0 and then to prove I

that {5;} is w-w-relatively compact. Shift attention to the matching measures f.1i on the Stone-tech compactification Q(S) and let f.1= L 1f.1;i/2 i. iew

Then each f.1i is absolutely continuous with respect to f.1, so we can shift attention again to matching fi in LI (f.1). If {5;} is not w-w-compact in m(S)*, then {Ii} is not w-w-compact in LI (f.1); use the criterion of III, 2, (9 a), to see that there exist 8> 0, measurable sets E j , and indices i j such that Sfijdf.1?88 for all j in w. Subsequencing on j, either

Sh + d f.1? 4 8 or Sh - d f.1 ? 4 8; the first case is typical. Then there Ej

Ej

exist open-and closed sets Vj 2. E j such that If.1i) (Vj \E) < 2 Ii so f.1ij(V)?28. But limf.1i(V)=O since XujEC(Q(S)); by subsequencing J

again and setting VI = VI' V z = Vjz\VI' etc., we get f.1i/V)?'8 and Vi disjoint open-and-closed subsets of Q(S). Then the map T of m(w) into C(Q(S)) for which Tx=x n in V n , Tx=O outside the closure of U V n , and Tx is defined by continuity elsewhere in Q(S), can be used nEW

in Cor. 11,2,1 to say that 8:Sf.1iJVn ):s

L If.1iJV)I--+O,

a contradiction

JEW

which tells us that {5J (and even {5t}) is a w-w-relatively compact

§ 4. Special Properties of (AL)-Spaces

141

set. But each w-convergent subsequence of (Ei) is also w*-convergent to 0, so (Ei) is w-convergent to 0.) (b) A generalization and consequence of (a) is also due to Grothendieck [5]. If S is compact Hausdorff and extremally disconnected, then every w*-convergent sequence in C(S)* is w-convergent to the same limit.

(By Theorem V, 4,3, when C(S) is represented as a subspace of m(U") there is a projection P of norm one of m(U") on C(S). p* carries w*convergent sequences in C(S)* to w*-convergent sequences in m(U")*.) (c) The same proof shows that if B is a Banach space in the family ~ (Def. V,4, 1), then w*- and w-sequential convergence are equivalent in B*. (d) Schaefer [2], among others, has polished Grothendieck's proof to carry through for a C(S) with S quasi-Stonean, that is, the closure of every open Fa-subset of S is open. This property of S is equivalent to: Every countable bounded set in C(S) has a least upper bound.

Chapter VII. Metric Geometry in Normecl Spaces

§ 1. Isometry and the Linear Structure Banach's book, p. 160, gives a theorem of Mazur and Ulam that an isometry of one normed space onto another which carries 0 to 0 is linear. This is true only for real-linear spaces, and is proved by characterizing the midpoint of a segment in a normed space in terms of the distance function. Using the same proof a slightly stronger result can be attained. Theorem 1. In a locally convex linear topological space over the real field the uniformity and the zero point determine the linear structure.

Most of the proof depends on Lemma 1. Let L be an LeS and let {p.: seS} be a family of continuous pre-norms large enough to separate points of L; for example, this might be the family of all continuous pre-norms in L. For each s let d. be defined by d.(x,y) = P.(x - y). Then for each Xl' X2 in L the midpoint XO=(XI +x 2)/2 of the segment from Xl to X2 can be found in terms of the set {d.: s e S} of pre-metrics.

Proof. To save subscripts, temporarily let d be anyone of the d.; then define El ={x:xeL and d(Xl'X) = d(x,x 2) =d(xl ,x2)/2}. If En is defined, let D(EJ be the diameter of En measured in terms of the premetric d; then set En+l =Enn{x:d(x,y)~D(En)/2 for every y in En}. It follows easily that if En+ 1 is not empty, then D(En+ 1) ~ D(EJ/2. Next it is necessary to show that xoenEn; to this end define T by n

the formula Tx = Xl + X2 - X for each X in L. Then T is an isometry whose only fixed point is Xo, and xOeE l . To prove by induction on n that T(EJ~En for all n, take X in El ; then d(Tx,xJ=d(x l +x 2 -x,x j ) =d(x 3 _ j ,x), so TxeEl . Next suppose y and Ty in En and X in En+l> then d(Tx, y)

= d(Xl + X2 -

x,y) = d(Xl +X2 - y, x) = d(x, Ty) ~ D(En)/2,

§ 1. Isometry and the Linear Structure

so TXEEn+ I ' and induction proves

143 T(En)~En

for all n. Then for x in En

D(En) d( X,X o ) -_ d( Tx,xo ) -_ d(x,2Tx) < = 2 '

so xoEEn+1 if XoEEn. This proves xoEnEn=E. n

Now consider again the set of all {d s : SE S}. By the preceding argument each ds determines a set ES in which Xo lies; also the ds-diameter, Ds(£S), is zero; that is, if x E ES , then ds(x, x o) =0 for all s in S. But it was

n

SES

assumed to begin with that the family of pre-metrics was large enough to separate points; hence Xo is the only element of £S.

n

SES

The theorem will be proved if we add to the lemma the following special case of the theorem. Corollary 1. Call a one-to-one function T between two locally convex spaces tl unimorphy if it carries a separating family {d s: SES} of premetrics on L to another such family {d~: SE S} on L by the rule d~(Tx, Ty) = ds(x,y). Then each unimorphy T is affine. Proof. Given XI and x 2 in L, T carries the set £S (of the lemma) to the corresponding set E's in L constructed in a similar manner from Tx\ and Tx 2 • Hence (XI + x 2 )/2, the only point in all the £S, is carried by T to (TxI + Tx 2 )/2, the only point in all of the E's; this asserts that T preserves midpoints. As in the proof of Lemma 1,4, 1 it can now be proved that T(rx l +(1-r)x 2 )=rTx l +(1-r)Tx 2 if r is dyadic rational. The families {d s : SES} and {d~: SES} determine locally convex topologies in Land L, respectively, in which T is continous; hence, the above relation holds for all real r, and T is affine. This completes the proof of the corollary. If also TO =0, then T(rx)=T(rx+0)=T(rx+(1-r)0)=rTx; hence T is linear. This proofs applies, of course, to the normed case, where one norm is a separating family of continuous pre-norms. Charzynski has shown that an isometry of a finite-dimensional linear metric space which leaves 0 invariant must be linear. His proof consists of constructing from the given metric a pre-norm which is not identically zero and which is invariant under every isometry which has 0 as a fixed point. The proof then uses the Mazur-Ulam result and induction on the dimension of the space. Mankiewicz gives a generalization of the Mazur-Ulam theorem: Let V be an open connected subset of a normed space N and let T be an isometry of V onto an open subset of normed space N'; then T can be extended (uniquely) to an affine isometry of N onto N'. [The proof begins by proving T affine in some ball about each point of V; connect-

Chapter VII. Metric Geometry in Normed Spaces

144

edness makes consistent extension possible. A simple example in a twodimensional P space shows that connectedness of V is needed.]

§ 2. Rotundity and Smoothness If a convex set K has interior points, then it may happen that no open segment in K contains a boundary point of K. Such a set has been called "strictly convex" in many papers; here it shall be called rotund. To simplify the discussion it shall be assumed that the convex body is also bounded and symmetric about 0; then it may be taken to be the unit ball of a normed linear space. In this section N will be a normed space, B its completion, U the unit ball, L the surface of U = {x: I x I = 1}, un the unit ball of B* = N*, and L' the surface of un. Much of what follows can be found in Smul'yan [3,6,7], Klee [6], or Day [1 to 5J. Definition 1. N, or U, is rotund (R) if every open segment in U is disjoint from L. N, or U, is smooth (S) if at each point of L there is only one supporting hyperplane of U. (1) There are a number of other properties of N easily shown to be equivalent to (R): (R 1 ) Every point of L is an extreme point of U. (R 2 ) Every point of L is an exposed point of U. (R3) Every hyperplane of support of U touches L in at most one point (Ruston [3]). (R4) Supporting hyperplanes to U at distinct points of L are distinct. (Rs) If W is a wedge in N in which norm is additive, then W is a halfray {t x: t ~ O}. (KreIn calls such a U strictly normalized; he uses the property in work on the moment problem; see Ahiezer and KreIn.) (R6) In every twodimensional subspace L of N the unit ball L n U is rotund. (2) Properties equivalent to smoothness are: (SI) In every two-dimensional linear subspace L of N the unit ball L n U is smooth. [The proof that (SI) implies (S) requires the HahnBanach theorem.] (G) The norm functional in N has a Gateaux differential at each point of L; that is,

(i)

G(x,h)

= lim(llx+thll-llxll) 1-+0

t

exists for each x in Land h in N. (Because I I is a convex function, this implies that G(x, h) is a linear function of h.) (3) (a) If N* is (R) [or (S)], then N is (S) [or (R)]. (b) Hence, if B is reflexive, B is (R) [or (S)] if and only if B* is (S) [or (R)J. (c) If 11(W) is renormed as in 4, (2), it will be rotund, but 11(W)* is like m(w) which, by 4, (1), has no isomorphic smooth norm. Hence the duality of (b)

§ 2. Rotundity and Smoothness

145

requires reflexivity. (d) In the absence of reflexivity there is a substitute duality for going upstream: U" is (R) [or (S)] if and only if every twodimensional factor space of B is (S) [or (R)]. [This depends on the isometries described in Cor. II, 1, 1 and, of course, the two-dimensional case of (b). The uniformization of this fact is the crux of the argument of Day [4] sketched in (10a).] (e) Hence, by (a), for B to be (R) or (S) it suffices that every two-dimensional factor space of B have the same property. The example of (c) shows that rotundity of B need not imply rotundity of every factor space. Troyanski [3] has given a corresponding example for smoothness. Next we consider some functions, moduli of rotundity, which measure the 'minimum depth of a segment in U subject to various constraints. To say that one of these moduli for a space N is positive has a strong effect on the geometrical and other properties of Nand U. Definition 2. In the formulas below LI(x,y)=1-II(x+y)/211 is the depth of the midpoint of the segment [x,y] below 1: and the infimum is calculated subject to the general constraint that x and yare both in U.

c5(e) = inf{ LI(x,y): II x - yll :.:::e}. If Pis in 1:', then

c5(e,p) = inf{LI(x,y): IP(x- y)l:.:::e}. If x is in 1:, then

c5(x,e) = inf{LI (x, y): Ilx- yll :.:::e}. If x is in 1: and

Pin 1:', then c5(x, e, P) = inf{ LI (x,y): IP(x - y)l:.::: e}.

If z is in 1: then

c5(e, -H) = inf{ LI(X,X+AZ): IIAZII :.:::e}. Say that N, or U, is (UR) uniformly rotund if c5(e) >0 when 0s(IIP,+lll + IIP,II), then A(x,s) is a finite set, and (vi) if A(x) = U A(x,s) and Y x is the

closed linear hull of PweB)

+ U

£>0

(Pa+

1-

P~)(B), then

XE

Y x ; then B

'lEA (x)

has an isomorphic (LUR) norm. Proof. Let d" = {A: A is a set of not more than n ordinals a, w:::;;'a:::;;' 11, and let d = U d". For each a let (er, i E w) be a sequence fundamental nEW

in (P,+l-P,)(B), and let (e;,iEw) be a similar sequence in PwB. For each A in d and m in W define the following functions on B: E~(x) = distance from x to the linear hull of

Gn(x) = sup{E:(x)+nFA(x): AEd,,}

for each n in w.

Go(x) = Ilxll. Let S be a set for which an appropriate T maps B into cu(S) and let V= {O, -1, -2, ... } u {a: w:::;;.a from B into co(V) by: [rJ>x](-n)=2- nGn(x), [rJ>x](a)=t,(x), and [rJ>x](s) = Tx(s) for s in S. Define a new norm in B by Ilxll'= IIrJ>x'll where the norm used in co(V) is the (LUR) norm of (3 a). Take a sequence (x k ) in B with I xkll' = 1 = Ilxll'

§ 4. Isomorphisms to Improve the Norm

163

and Ilxk+xll'--+2; then IIcf>Xk-cf>xllco(v)--+O so ta(Xk)--+ta(x) for all ()(, Gn(Xk)--+Gn(x) for each n and II TXk - Txllco(s)--+O. This last condition implies that if a subsequence of (x k) converges, then it must converge to x, so the rest of the proof shows that (x k) must be norm-totallybounded. Given 8>0, choose m in wand A" in A so that El,"(x) k" 8

E:(x k) < Gn(X) -nFA(x k)< Gn(X) -nFA(x) +"3 < E:(x) +

28

3

< 8.

Hence (x k) is bounded and is within 8 of a finite dimensional subspace Ce of B; since each ball in C is norm-compact, (x k) is covered by a finite set of balls of radius 8. This shows that every subsequence of (x k) contains a convergent subsequence; but we saw that such a convergent subsequence must have limit x, so (x k ) converges to x. Corollary 4 (Troyan ski [2]). Every Banach space with a (transfinite) basis is O there is an n(r) in (J) such that no n-c-tree lies in U, the unit ball of N. (N y) N is not (y); in detail, there is 10 >0 such that for each n in w there is an n-s-tree in U [This is condition P 1 of James [7].] (Q) N is quadrate; that is, N mimics m 2 . (NQ) N is not (Q); in detail, there is a number a with 0 < a < 1 such that if Ilxll ~ 1"?c.llyll, then Ilx+ yll