Introductory Theory of Topological Vector Spaces 9780203749807, 0203749804, 9781351436465, 1351436465, 0-8247-8779-X

Table of contents :
Content: Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
1: Hahn-Banach's extension theorem
2: Banach spaces and Hilbert spaces
3: Operators and some consequences of Hahn-Banach's extension theorem
4: The uniform boundedness theorem and Banach's open mapping theorem
5: A structure theorem for compact sets in a Banach space
6: Topological injections and topological surjections
7: Vector topologies
8: Separation theorems and Krein-Milman's theorem
9: Projective topologies and inductive topologies
10: Normed spaces associated with a locally convex space 11: Bounded sets and compact sets in metrizable topological vector spaces12: The bornological space associated with a locally convex space
13: Vector bornologies
14: Initial bornologies and final bornologies
15: von Neumann bornologies and locally convex topologies determined by convex bornologies
16: Dual pairs and the weak topology
17: Elementary duality theory
18: Semi-reflexive spaces and ultra-semi-reflexive spaces
19: Some recent results on compact operators and weakly compact operators
20: Precompact seminorms and Schwartz spaces
21: Elementary Riesz-Schauder's theory 22: An introduction to operator ideals23: An aspect of fixed point theory
24: An introduction to ordered convex spaces
Special symbols
References
Index

Citation preview

INTRODUCTORY THEORY OF TOPOLOGICAL VECTOR SPACES

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft

Zuhair N ashed U niversity o f D e la w a re N ew ark, D e la w a re

R utgers U n iversity N ew B runsw ick, N ew J e rse y

CHAIRMEN OF THE EDITORIAL BOARD S.

K obayashi

U n iversity o f C alifornia, B erkeley B erkeley, C alifornia

Edwin H ew itt U niversity o f W ashington Seattle, W ashington

EDITORIAL BOARD M. S. B aouendi U n iversity o f C alifornia, San D ie g o Ja ck K. H ale G eorgia Institute o f T echnology M arvin M arcu s U n iversity o f C alifornia, Santa B a rb a ra W. S. M assey Yale U niversity L e o p o ld o N achbin C entro B rasileiro d e P esq u isa s F isicas A n il N erode C orn ell U n iversity

D o n ald P assm an U niversity o f W isconsin—M adison F red S. R oberts R utgers U niversity G ian-C arlo R ota M assachu setts Institute o f T echnology D a v id L. R ussell Virginia P olytechnic Institute an d State University Jane Cronin Scanlon R utgers U niversity W alter Schem pp U niversitat Siegen

M ark Teply U n iversity o f W isconsin—M ilw aukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS

1. K. Yano, Integral Formulas in Riemannian G eom etry (1 9 7 0 ) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1 9 7 0 ) 3 . V. S. Vladimirov, Equations of Mathem atical Physics (A. Jeffrey, ed.; A. Little wood, trans.) (1 9 7 0 ) 4 . B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation ed.; K. M akow ski, trans.) (1 9 7 1 ) 5. L. N arici e t a!., Functional Analysis and Valuation Theory (1 9 7 1 ) 6. S. S. Passman, Infinite Group Rings (1 9 7 1 ) 7 . L. Dornhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1 9 7 1 , 1 9 7 2 ) 8 . W. Boothby and G. L. Weiss, eds., Sym metric Spaces (1 9 7 2 ) 9 . Y. M atsushim a, Differentiable Manifolds (E. T. Kobayashi, trans.) (1 9 7 2 ) 1 0 . L £ Ward, Jr., Topology (1 9 7 2 ) 1 1 . A . Babakhanian, Cohomological Methods in Group Theory (1 9 7 2 ) 1 2 . R. Gilmer, M ultiplicative Ideal Theory (1 9 7 2 ) 1 3 . J. Yeh, Stochastic Processes and the W iener Integral (1 9 7 3 ) 1 4 . J. Barros-Neto, Introduction to the Theory of Distributions (1 9 7 3 ) 1 5 . R. Larsen, Functional Analysis (1 9 7 3 ) 16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles (1 9 7 3 ) 1 7 . C. Procesi, Rings w ith Polynomial Identities (1 9 7 3 ) 1 8 . R. H erm ann, Geom etry, Physics, and Systems (1 9 7 3 ) 1 9 . N. R. Wal/ach, Harmonic Analysis on Homogeneous Spaces (1 9 7 3 ) 2 0 . J. Dieudonne, Introduction to the Theory of Formal Groups (1 9 7 3 ) 2 1 . /. Vaisman, Cohomology and Differential Forms (1 9 7 3 ) 2 2 . B.-Y. Chen, G eom etry of Submanifolds (1 9 7 3 ) 2 3 . M . M arcus, Finite Dimensional Multilinear Algebra (in tw o parts) (1 9 7 3 , 1 9 7 5 ) 2 4 . R. Larsen, Banach Algebras (1 9 7 3 ) 2 5 . R. O. Kujala and A . L. Vitter, eds., Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1 9 7 3 ) 2 6 . K. B. Sto/arsky, Algebraic Numbers and Diophantine Approximation (1 9 7 4 ) 2 7 . A . R. M agid, The Separable Galois Theory of Com m utative Rings (1 9 7 4 ) 2 8 . B. R. M cDonald, Finite Rings with Identity (1 9 7 4 ) 2 9 . J. S atake, Linear Algebra (S. Koh et al., trans.) (1 9 7 5 ) 3 0 . J. S. Golan, Localization of Noncommutative Rings (1 9 7 5 ) 3 1 . G. Klambauer, Mathem atical Analysis (1 9 7 5 ) 3 2 . M . K. Agoston, Algebraic Topology (1 9 7 6 ) 3 3 . K. R. Goodearl, Ring Theory (1 9 7 6 ) 3 4 . L. £ M ansfield, Linear Algebra with Geometric Applications (1 9 7 6 ) 3 5 . N. J. Pullman, Matrix Theory and Its Applications (1 9 7 6 ) 3 6 . B. R. M cDonald, Geometric Algebra Over Local Rings (1 9 7 6 ) 3 7 . C. W. Groetsch, Generalized Inverses of Linear Operators (1 9 7 7 ) 3 8 . J. £ Kuczkow ski and J. L. Gersting, Abstract Algebra (1 9 7 7 ) 3 9 . C. O. Christenson and W. L. Voxman, Aspects of Topology (1 9 7 7 ) 4 0 . M . N agata, Field Theory (1 9 7 7 ) 4 1 . R. L. Long, Algebraic Number Theory (1 9 7 7 ) 4 2 . W. F. Pfeffer, Integrals and Measures (1 9 7 7 ) 4 3 . R. L. Wheeden and A . Zygmund, Measure and Integral (1 9 7 7 ) 4 4 . J. H. Curtiss, Introduction to Functions of a Complex Variable (1 9 7 8 ) 4 5 . K. Hrbacek and T. Jech, Introduction to Set Theory (1 9 7 8 ) 4 6 . W. S. M assey, Homology and Cohomology Theory (1 9 7 8 ) 4 7 . M . M arcus, Introduction to Modern Algebra (1 9 7 8 ) 4 8 . £ C. Young, Vector and Tensor Analysis (1 9 7 8 ) 4 9 . S. B. Nadler, Jr., Hyperspaces of Sets (1 9 7 8 ) 5 0 . S. K. Segal, Topics in Group Kings (1 9 7 8 ) 5 1 . A . C. M . van Rooij, Non-Archimedean Functional Analysis (1 9 7 8 ) 5 2 . L. Corwin and R. Szczarba, Calculus in Vector Spaces (1 9 7 9 )

53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107.

C. Sadosky, Interpolation of Operators and Singular Integrals (1 9 7 9 ) J. Cronin, Differential Equations (1 9 8 0 ) C. W. Groetsch, Elements of Applicable Functional Analysis (1 9 8 0 ) /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1 9 8 0 ) H. /. Freedan, Deterministic M athem atical Models in Population Ecology (1 9 8 0 ) S. B. Chae, Lebesgue Integration (1 9 8 0 ) C. S. Rees et al., Theory and Applications of Fourier Analysis (1 9 8 1 ) L. Nachbin, Introduction to Functional Analysis (R. M . Aron, trans.) (1 9 8 1 ) G. Orzech and M. Orzech, Plane Algebraic Curves (1 9 8 1 ) R. Johnsonbaugh and W. E Pfaffenberger, Foundations of M athem atical Analysis (1 9 8 1 ) W. L. Voxman and R. H. Goetschel, Advanced Calculus (1 9 8 1 ) L. J. Corwin and R. H. Szcarba, Multivariable Calculus (1 9 8 2 ) V. /. Istratescu, Introduction to Linear Operator Theory (1 9 8 1 ) R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces (1 9 8 1 ) J. K. Beem and P. E. Ehrlich, Global Lorentzian Geom etry (1 9 8 1 ) D. L. A rm acost, The Structure of Locally Compact Abelian Groups (1 9 8 1 ) J. W. Brew er and M. K. Sm ith, eds., Emily Noether: A Tribute (1 9 8 1 ) K. H. Kim, Boolean M atrix Theory and Applications (1 9 8 2 ) T. W. Wieting, The M athem atical Theory of Chromatic Plane Ornaments (1 9 8 2 ) D. B.Gau/d, Differential Topology (1 9 8 2 ) R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1 9 8 3 ) M. Carmeli, Statistical Theory and Random Matrices (1 9 8 3 ) J. H. Carruth et al., The Theory of Topological Semigroups (19 8 3 ) R. L. Faber, Differential G eom etry and Relativity Theory (1 9 8 3 ) S. Barnett, Polynomials and Linear Control Systems (1 9 8 3 ) G. Karpi/ovsky, Com m utative Group Algebras (1 9 8 3 ) F. Van Oystaeyen and A . Verschoren, Relative Invariants of Rings (1 9 8 3 ) /. Vaisman, A First Course in Differential Geom etry (1 9 8 4 ) G. W. Sw an, Applications of Optimal Control Theory in Biomedicine (1 9 8 4 ) T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1 9 8 4 ) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1 9 8 4 ) T. Albu and C. Nastasescu, Relative Finiteness in Module Theory (1 9 8 4 ) K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (1 9 8 4 ) F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1 9 8 4 ) B. R. M cDonald, Linear Algebra Over Comm utative Rings (1 9 8 4 ) M. Namba, G eom etry of Projective Algebraic Curves (1 9 8 4 ) G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1 9 8 5 ) M. R. Bremner e t a/., Tables of Dominant W eight Multiplicities forRepresentations of Simple Lie Algebras (1 9 8 5 ) A. E. Fekete, Real Linear Algebra (1 9 8 5 ) S. B. Chae, Holomorphy and Calculus in Normed Spaces (1 9 8 5 ) A. J. Jerri, Introduction to Integral Equations with Applications (1 9 8 5 ) G. Karpi/ovsky, Projective Representations of Finite Groups (19 8 5 ) L. N arici and £. Beckenstein, Topological Vector Spaces (1 9 8 5 ) J. Weeks, The Shape of Space (1 9 8 5 ) P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research (1 9 8 5 ) J .-A . Chao and W. A . Woyczynski, eds., Probability Theory and Harmonic Analysis (1 9 8 6 ) G. D. Crown et at., Abstract Algebra (1 9 8 6 ) J. H. Carruth et a!., The Theory of Topological Semigroups, Volume 2 (1 9 8 6 ) R. S. Doran and V. A . Belfi, Characterizations of C *-Algebras (1 9 8 6 ) M. W. Jeter, M athem atical Programming (1 9 8 6 ) M. A ltm an, A Unified Theory of Nonlinear Operator and Evolution Equations w ith Applications (1 9 8 6 ) A. Verschoren, Relative Invariants of Sheaves (1 9 8 7 ) R. A . Usmani, Applied Linear Algebra (1 9 8 7 ) P. Blass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1 9 8 7 ) J. A . Reneke et a!., Structured Hereditary Systems (1 9 8 7 )

1 0 8 . H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesics (1 9 8 7 ) 1 0 9 . R. H a rte , Invertibility and Singularity for Bounded Linear Operators (1 9 8 8 ) 1 1 0 . G. S. Ladde et a!., Oscillation Theory of Differential Equations w ith Deviating Argum ents (1 9 8 7 ) 1 1 1 . L. Dudkin et a!., Iterative Aggregation Theory (1 9 8 7 ) 1 1 2 . T . Okubo, D ifferential Geometry (1 9 8 7 ) 1 1 3 . D. L. S ta n d and M . L. S tand, Real Analysis with Point-Set Topology (1 9 8 7 ) 1 1 4 . T. C. Gard, Introduction to Stochastic Differential Equations (1 9 8 8 ) 1 1 5 . S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1 9 8 8 ) 1 1 6 . H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations (1 9 8 8 ) 1 1 7 . J. A. Huckaba, Commutative Rings with Zero Divisors (1 9 8 8 ) 1 1 8 . W. D. Wallis, Combinatorial Designs (1 9 8 8 ) 1 1 9 . W. Wiqstaw, Topological Fields (1 9 8 8 ) 1 2 0 . G. Karpilovsky, Field Theory (1 9 8 8 ) 1 2 1 . S. Caenepeei and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1 9 8 9 ) 1 2 2 . W. Koz/ow ski, Modular Function Spaces (1 9 8 8 ) 1 2 3 . E. Lowen-Colebunders, Function Classes of Cauchy Continuous M aps (1 9 8 9 ) 1 2 4 . M . Pavel, Fundamentals of Pattern Recognition (1 9 8 9 ) 1 2 5 . V. Lakshmikantham e t a!., Stability Analysis of Nonlinear Systems (1 9 8 9 ) 1 2 6 . R. Sivaramakrishnan, The Classical Theory of Arithm etic Functions (1 9 8 9 ) 1 2 7 . N. A . Watson, Parabolic Equations on an Infinite Strip (1 9 8 9 ) 1 2 8 . K. J. Hastings, Introduction to the M athem atics of Operations Research (1 9 8 9 ) 1 2 9 . B. Fine, Algebraic Theory of the Bianchi Groups (1 9 8 9 ) 1 3 0 . D. N. Dikranjan et al., Topological Groups (1 9 8 9 ) 1 3 1 . J. C. M organ II, Point Set Theory (1 9 9 0 ) 1 3 2 . P. Bi/er and A . W itkowski, Problems in Mathem atical Analysis (1 9 9 0 ) 1 3 3 . H. J. Sussmann, Nonlinear Controllability and Optimal Control (1 9 9 0 ) 1 3 4 . J.-P. Florens et al., Elements of Bayesian Statistics (1 9 9 0 ) 1 3 5 . N. Shell, Topological Fields and Near Valuations (1 9 9 0 ) 1 3 6 . B. F. Doolin and C. F. M artin, Introduction to Differential G eom etry for Engineers (1 9 9 0 ) 1 3 7 . S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1 9 9 0 ) 1 3 8 . J. Oknihski, Semigroup Algebras (1 9 9 0 ) 1 3 9 . K. Zhu, Operator Theory in Function Spaces (1 9 9 0 ) 1 4 0 . G. B. Price, An Introduction to Multicomplex Spaces and Functions (1 9 9 1 ) 1 4 1 . R. B. Darst, Introduction to Linear Programming (1 9 9 1 ) 1 4 2 . P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1 9 9 1 ) 1 4 3 . T. Husain, Orthogonal Schauder Bases (1 9 9 1 ) 1 4 4 . J. Foran, Fundamentals of Real Analysis (1 9 9 1 ) 1 4 5 . W. C. Brown, M atrices and Vector Spaces (1 9 9 1 ) 1 4 6 . M. M . Rao and Z. D. Ren, Theory of Orlicz Spaces (1 9 9 1 ) 1 4 7 . J. S. Golan and T. Head, Modules and the Structures of Rings (1 9 9 1 ) 1 4 8 . C. Small, Arithmetic of Finite Fields (1 9 9 1 ) 1 4 9 . K. Yang, Complex Algebraic Geometry (1 9 9 1 ) 1 5 0 . D. G. H offm an et a!.. Coding Theory (1 9 9 1 ) 1 5 1 . M. O. Gonzalez, Classical Complex Analysis (1 9 9 2 ) 1 5 2 . M. O. Gonzalez, Complex Analysis (1 9 9 2 ) 1 5 3 . L. W. Baggett, Functional Analysis (1 9 9 2 ) 1 5 4 . M. Sniedovich, Dynamic Programming (1 9 9 2 ) 1 5 5 . R. P. A g arw al, Difference Equations and Inequalities (1 9 9 2 ) 1 5 6 . C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1 9 9 2 ) 1 5 7 . C. S w artz, An Introduction to Functional Analysis (1 9 9 2 ) 1 5 8 . S. B. Nadler, Jr., Continuum Theory (1 9 9 2 ) 1 5 9 . M. A . A l-G w aiz, Theory of Distributions (1 9 9 2 ) 1 6 0 . £ Perry, Geometry: Axiomatic Developments with Problem Solving (1 9 9 2 ) 1 6 1 . £. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1 9 9 2 ) 1 6 2 . A . J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1 9 9 2 ) 1 6 3 . A . Charlier e t a!., Tensors and the Clifford Algebra (1 9 9 2 )

1 6 4 . P. Biler and T. Nadzieja, Problems and Examples in Differential Equations (1 9 9 2 ) 1 6 5 . £. Hansen, Global Optimization Using Interval Analysis (1 9 9 2 ) 1 6 6 . S. Guerre-Delabriere, Classical Sequences in Banach Spaces (1 9 9 2 ) 1 6 7 . / . C. Wong, Introductory Theory of Topological Vector Spaces (1 9 9 2 ) 1 6 8 . S. H. Ku/karni and B. V. Limaye, Real Function Algebras (1 9 9 2 )

Additional Volumes in Preparation

INTRODUCTORY THEORY OF TOPOLOGICAL VECTOR SPACES

Yau-Chuen Wong Departm ent o f Mathematics The Chinese University o f Hong Kong Shantin NT, Hong Kong

Marcel Dekker, Inc.

New York • Basel • Hong Kong

Library of Congress Cataloging-in-Publication Data Wong, Yau-Chuen. Introductory theory of topological vector spaces / Yau-Chuen Wong, p. cm. — (Monographs and textbooks in pure and applied mathematics ; 167) Includes bibliographical references (p. ) and index. ISBN 0-8247-8779-X (acid-free paper) 1. Linear topological spaces. I. Title. II. Series. QA322.W66 1992 5 1 5 \7 3 —dc20 92-20435 CIP

This book is printed on acid-free paper.

Copyright ©

1992 by MARCEL DEKKER, INC. All Rights Reserved.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, micro­ filming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

To My C hildren

K A R EN K A -W IN G and JA SO N C H IT —KA Y

Preface

T his elem en tary te x t aim s to present concisely th e basic concepts and principles of fun ctio n al

analysis,

as

well as its

applications,

to

first—year g ra d u a te

and

senior

u n d e rg ra d u a te stu d e n ts in m athem atics. It includes som e recent results th a t can be tau g h t a t an elem en tary level and enable stu d en ts to reach quickly th e fro n tiers of current m a th e m a tic a l research. N um erous helpful exercises are also included. T he first six chapters of this te x t are concerned w ith th ree fu n d a m e n ta l results (the H ah n —B anach extension theorem , B anach's open m apping theorem an d th e uniform boundedness theorem ) on Banach spaces, together w ith G ro th en d ieck 's stru c tu re theorem for com pact sets in B anach spaces, and Helley's selection theorem as well. As th e unit ball in B an ach spaces is a neighborhood of 0 as well as a bounded set, to e x te n d th e concept of th e u n it ball to m ore general topological vector spaces, it is n a tu ra l to stu d y parallelly vector topologies and vector bornologies, which are included respectively in C hapters 7~11 a nd C h a p te rs 12~15. C hapters 16~18 concentrate on th e in te rn a l and e x te rn al d u ality betw een bornologies and locally convex topologies. C hapters 19~22 are devoted to a study of som e recent results on com pact and weakly com pact op erato rs, o p e rato r ideals, and th eir a pplications to th e im p o rta n t class of Schw artz spaces. T h e tw o final chapters deal respectively w ith som e applications to fixed point theory, sta rtin g from Ky F a n 's K K M coverings (C h a p te r 23 was w ritten by Professor Shih M au—H siang, C hung—Y uan C hristian U niversity a t T aiw an), and d u ality theory on ordered vector spaces to g eth e r w ith th e c o n tin u ity of positive linear m appings. W ith regard to th e selection of the m aterials for th is book, m ore w ith Professor W . Rudin :

I cannot agree

"In order to w rite a book of m o d era te size, it was

therefore necessary to select certain areas and to ignore others.

I fully realize th a t alm ost

any expert who looks at th e tab le of contents will find th a t some of his (and my) favorite topics are m issing, b u t th is seems unavoidable. It was not m y inten tio n to w rite an encyclopedic trea tise. I w an ted to w rite a book th a t would open the way to fu rth er exploration" (quoted from his book F unctional Analysis (T a ta M c G ra w -H ill,1973)). The list of references at th e end of th is book is not a bibliography; it contains only books and articles to which we have m ade reference in the tex t. W e have m ade no system atic a tte m p t to a ttrib u te theorem s to th eir authors, nor have given any references to the original research p aper in which they appeared. T his book has been used successfully a t the Chinese U niversity of Hong Kong since 1978, and was also used as a te x t for a

one—sem ester course in functional analysis, at th e

g ra d u ate level, at C hung—Y uan C h ristian U niversity during the academ ic year 1987—88. T he m aterial of this book will be useful to g ra d u ate stu d en ts in analysis, as well as research workers in the field. It is hoped th a t stu d e n ts in m ath em atical physics, m ath em atical econom ics, or engineering will also profit from th e presentation. I would like to th an k th e N ational Science Council of the R epublic of C hina for financial su p p o rt during my sta y in T aiw an. I am especially grateful to C hung—Y uan C hristian U niversity, and U nited College of th e Chinese U niversity of Hong Kong for both financial and m oral su p p o rt.

T he friendly atm osphere and the working environm ent at

C hung—Y uan were conducive to scientific research. I wish to express my deep g ra titu d e to Professors Ky Fan, who encouraged me to w rite this book and suggested m any helpful hints in the tex t. Also, I would like to th an k Professors Shih M au—Hsiang and T u Shih—Tong (C hung—Y uan), Chen M ing—Po and C hiang T zuu—Shuh (In s titu te of M athem atics, Academ ic Sinica at Taiw an) for stim u latin g discussions and valuable suggestions; special thanks go to Professor A .J.E llis (U niversity of Hong Kong) for his valuable com m ents. My appreciation is extended to Professor Shih M au—Hsiang for the lectures he gave th ere and for allowing me to include his notes in C h a p te r 23.

It is a pleasure to acknowledge m y indebtedness to m any friends, colleagues and stu d e n ts for th eir help. Special thanks are due to Miss C hiang Y ueh—hwa, Mr. W en C hyi—yau, M r. Fan Cheng—San Andrew, and Mr. C heung M ing—W ai for th e ir typing of the m an u scrip t. Finally, I wish to express m y appreciation to th e editors and staff of Marcel D ekker for th eir effective cooperation.

Y au—C huen W ong

Contents P reface

v

1.

H ahn—B anach's extension theorem

2.

B anach spaces and H ilbert spaces

13

3.

O perators and some consequences of H ahn—B anach's extension theorem

47

4.

T he uniform boundedness theorem and B anach's open m apping theorem

75

5.

A stru c tu re theorem for com pact sets in a B anach space

91

6.

T opological injections and topological surjections

98

7.

V ector topologies

110

8.

S eparation theorem s and K rein—M ilm an's theorem

132

9.

P rojective topologies and inductive topologies

149

10.

N orm ed spaces associated w ith a locally convex space

162

11.

B ounded sets and com pact sets in m etrizable topological vector spaces

170

12.

T he bornological space associated w ith a locally convex space

175

13.

V ector bornologies

180

14.

In itia l bornologies and final bornologies

190

15.

von N eum ann bornologies and locally convex topologies d eterm ined by convex bornologies

198

16.

D ual pairs and th e weak topology

205

17.

E lem entary d uality theory

216

18.

Semi—reflexive spaces and u ltra —semi—reflexive spaces

238

19.

Some recent results on com pact operators and w eakly com pact operators

246

20.

P recom pact sem inorm s and Schw artz spaces

265

21.

E lem en tary Riesz—Schauder's theory

282

22.

An in tro d u ctio n to operator ideals

297

1

ix

23.

An aspect of fixed point theory

331

24.

An in tro d u ctio n to ordered convex spaces

353

Special sym bols

387

References

395

Index

411

1

Hahn-Banach’s Extension Theorem T h roughout these lecture notes, the le tte r K will d enote e ith e r th e real field IR or th e com plex field C . V ector spaces m ean vector spaces over th e field K ; vector spaces over IR (resp. C) are called real (resp. com plex) vector spaces. F or any subsets

A

and

B of a

v ector space E and A, // G K , we define

AA T fiB — { Aa T //b ; a T he expression

{x} + A

6

A , b G B }.

will be a bbreviated by

x + A (x e E ), (—1 )A

by

—A

and A + (—B) by A — B. Let E be a vector space and B C E . Recall th a t B is: (a) convex if AB + (1 — A)B C B whenever A G [0, 1]; (b) sym m etric if B = —B; (c) circled (or balanced) if AB C B whenever

| A| < 1;

(d) absolutely convex (or disked, disk) if it is convex and circled; (e) absorbing (or radical at

x e /jB

0 ) if

for all / i g K w ith

for any x GE ,th ere exists an

A>

0

such th a t

|//| > A.

T he intersection of a family of convex (resp. circled, disked) sets is convex (resp. circled, disked), hence th e convex hull (resp. circled h u ll, disked h u ll) of a given set denoted circled,

by

co(B) (resp. ch(B ), TB), is defined to be th e in te rse ctio n of all convex

B , (resp.

disked) sets c o n ta in in g B. If B = U Bj , th en we w rite co(B j) (resp. ch(B j), jeA j j

TBj) for co(B) (resp. ch(B ), TB). It is

easily seen th a t

co(B) = { E n^A x : x G B, A. > 0 and E / ^

ch(B ) =

U

AB ;

|A| 0).

A sublinear functional p on E is a sem inorm if (SN3) (H om ogeneity) p(/ix) = |/x |p (x )

(for all x G E and fi e K ).

Let p be a sublinear functional on E . For any A >

0

o | , A^ and and

, let

(A) = o ( A) = { x e E : p(x) < A } and

= {x

T hen

0

u |F

6

E : p(x) < A } = F F

are convex and absorbing; if in addition, p

is a sem inorm th en

l j |, A^ are c irc le d .lt is of in te rest to know w hether given an absorbing, disked

subset V of E , th ere is som e sem inorm p on E for which

e ith er V = O ' 1* or V = U 'n . E

E

W e shall see th a t this is a bit too m uch ask for. N evertheness one can get reasonably closed to such a result. T o do th is, we require th e following:

(1.1)

D efinition. Let V be an absorbing subset of E . F or each x e E , let

P v (x)

= inf{ A > 0 :

x

e AV }.

3

T h en p y is called th e gauge (or th e Minkowski fun ctio n al) of V.

R e m a rk . T he fact th a t V is absorbing ensures th a t p y (x) is fin ite for any x

(1.2)

6

E.

L em m a. Let V be an absorbing subset of E a n d p y th e gauge of V.

(a) If V is convex, then py is a sublinear fu n c tio n al. (b) If V is a d isk , th en p y is a sem inorm w ith th e p ro p e rty th a t

{x 6 E : p v (x) < 1 } C V C {

(i)

X

e E : p y(x) < 1 }.

M oreover, if q is a sem inorm on E such th a t

(2 )

{ x € E : q(x) < 1 } C V C {

x

G E : q(x) < 1 },

th e n p y = q .

P ro o f, (a) T he positive hom ogeneity of p y is obvious. T o prove th e su b a d d itiv ity , let A > 0 and // >

0

be such th a t

x g AV and y e f N . T h en th e convexity of V ensures th a t x + y

A

so th a t p y (x + y) < A + (i\ consequently,

p v ( x + y ) < P v (x ) + p v (y )

since A a n d /* were a rb itrary .

4

(b) If V is circled th e n we have

P v (//x) =

(for all f i e K and x c E).

|/ x |p v (x)

T o prove ( 1 ), let x e E be such th a t p y (x) < l.B y th e definition of infim um , th ere is an A w ith

0

< A
1 Show th a t th e convex hull of a circled set is circled, and th a t (by an exam ple) th e cik(V ) = {xeV : Ax£V for all | A |

(f)

| A-1 F be a linear m ap (i.e. T (A x+ ^y) = ATx + //Ty (w henever \ ,/ i £ K a n d x,y £ E )). Show th a t th e im age of a set W c E th a t is resp. convex, circled, disked, u nder T has th e sam e corresponding property.

1- 2 .

P rove (l.a ).

1-3.

P rove (l.b ).

1- 1.

P rove (l.c ).

1 -5.

Let E and F be vector spaces, let T :E —>F be linear and M a vector subspace of E. W e say th a t T is co m p atib le w ith M if M £ K er T = {x£E : T x= 0} (the kernel of T ).

(a)

Suppose th a t T is com patible w ith M. Show th a t th ere exists a unique linear m ap T :

'F such th a t T = T Q M , w here Q M : E - ^ E / M is th e q u o tie n t m ap (T is

called th e m ap o b tain e d from T by passing to the q u o tie n t.)

11

(b)

Show th a t th ere exists a unique bijective linear m ap ^ : E /j£ er

—> Im T (called

th e biiection associated w ith T ) such th a t T = J ^ Q r p w here Qrp : E— E / Ker ^ is th e q u o tien t m ap an d Jrp : Im T —>F is th e canonical em bedding. (T h e m ap

=

T is called th e injection associated w ith T .) 1 —6 .

Let E, F and G

(a)

(Induced m ap from the left). G iven a linear m ap

( a .l)

be vector spaces and T : E—>F a linear m ap. S: E —>G and suppose th a t

S is surjective (i.e. onto) and Ker S c K er T.

Show th a t th ere exists a unique linear m ap L : G —>F such th a t

(a . 2 )

T = LS.

M oreover, L is injective (i.e. one—one) if and only if

(a.3) (b)

Ker T c Ker S. (i.e. Ker T = K er S).

(Induced m ap from th e right) G iven a linear m ap S : G — >F a n d suppose th a t

(b .l)

S is injective and Im T c Im S.

Show th a t th ere exists a unique linear m ap R : E —> G such th a t

(b.2)

T = SR.

M oreover, R is surjective if and only if

Im S c Im T (i.e. Im T = Im S). N o te . T his p a rt is still true for sets ; in this case, all linear m aps a re only m appings. (1—7)

(A generalization of H ahn—B anach's extension th eo rem (A ndenaes[1970])). Let E be a real vector space, let p be a sublinear functional on E, let M be a vecto r subspace *

of E and let g e M

(a linear functional on M) be such th a t g(z) < p(z) for all z e M.

12

Show th a t for any B c E, th ere exists an f e E (i) (ii)

w ith the following properties:

f(z) = g(z) (for all z 6 M) and f(x) < p(x) (x e E); * if h e E , satisfied (i), is such th a t f | B - h IB ’ then f = h on B; [in o th er w ords, f is * a m axim al elem ents in th e set r(g,p) = {feE : f= g on M and f

0

such th a t

/*p(x) < q(T x) < Ap(x)

(for all x e E).

0

and fi >

15

(2.a) T h e n o rm -to p o lo g y : Let (E ,|| • ||) be a norm ed space. T h en th e fam ily

( 2 -a .l)

{

consisting of absolutely

0

E :n >

1

}

(or { - U - U g : n >

1

}),

convex and absorbing se ts, is a local base a t

0

for

|| *||E—top;

m oreover, th e || • || —■ t op has th e following rem arkable properties: (i) T h e || • ||E—top is com patible w ith th e vector space o p e ratio n , i.e., th e m aps

( x , y ) -------►x + y : E x E

►E

and (A ,x )-------►Ax : Kx E

►E

are continuous. (O f course, we consider th e product topology on th e p ro d u c t spaces.)

(ii) F or any x^ e E and 0 /

7q

e K, th e tran sla tio n , defined by

y -------►x Q + 70y

(for all y e E),

is a hom eom orphism from E onto E ; consequently, th e fam ily

( 2 .a. 2 )

{ XqH

— Ug : n >

1

}

(or { Xq H— O g : n >

1

})

is a local base a t x^ for th e || *||-^—top. As th e tran sla tio n and m ultiplication by non—zero scalars are hom eom orphism s (see ( 2 .a) (ii)), it follows th a t

x + A = x + A , XA = AA and w henever

x-e E, A e K and

A, B c E (where

A + H c (A +

A is th e closure of

B) A); m oreover, we have

th e follow ing : (2.2)

Proposition. Let E be a norm ed space and A, B c E .

16

00

(a )X =

00

-I

1

n (A + - i - 0 E ) = n (A + - 1 - U E ). n=l n=l

(b) A 4 - G is open w henever G is o p en , hence

A + IntB c In t(A

(c)

Let

K

+ B)

(w here Int B denotes th e interior of B).

c E be com pact and let

t hen th ere exists som e m >

(2.2.1)

1

B

be closed.

I|

K n B = (|) ,

such th a t

( K + - 1 - U E ) fl (B + - i - U E) = (|) .

C onsequently, if C c E is

Proof, (a) As - -

com pact and B is closed, th en C + B is closed

Ug c—

^

00

in E .

0 ^ c —jjj— U g , it follows th a t

1

|

00

n/ A + ^ ° E ) =

n=l

n

n=l

(A + ^

Ue ) '

Now p a rt (a) follows from th e following com putation: x e A

( x

H— — n

cj

) fl A / (b (for all n > 1 )

for a n y n > 1, th e re is som e a n G A such t h a t

x e a

&

(b )

For any Xq

it follows th a t

xn + G u

g

n

H— — O r (since 0 ^ = - Or-A n a, v n, iv

x G AH— i — O g (for all

n >

1 ).

E, th e tran sla tio n y ------ ►Xq + y (y 6 E) is a hom eom orphism ,

is open, and hence from

A + G =

U (a + G) aeA

th a t

A + G

is

open. Finally, since A + Int B c Int (A + B).

A + In t B

is an open subset of

A +

B , it follows th a t

17

(c) For any som e in teg er n(x) >

x G K, the closedness of B and 1

(x +

B n K = $ ensure th a t there exists

such th a t

n (x )

UE +

n (x )

UE +

n (x )

UE ) n

8

= 4> ,

and surely

(2.2.2)

(x + " n (x )

C learly

UE +

n (x )

U E* n

1

and i\j are ratio n al }.

T hen C is countable as well as dense in E , hence E is separable.

Remark. From th e preceding result, it is n atu ral to ask the following question :

(Q) Does every separable B anach space have a Schauder basis?

T his is a fam ous question raised by B anach m ore th an fifty years ago. Because of alm ost all known separable B anach spaces

had been shown to possess a Schauder basis, a positive

answer was expected for a long tim e. G rothendieck m ade a deep analysis of th is problem ; he found m any equivalent form ulations and consequences but no solution; he conjectured a negative

answ er.

In

1973,

Enflo

[1973]

succeeded

reflexive B anach space w hich has no Schauder

basis:

in

constructing a se p arab le,

his

ingenious but

highly

com plicated m ethods were sim plified to some degree by Davie [1973]. By a subspace of a norm ed space E is m eant a vector

subspace M of E ; while a

subspace of a B—space E is m eant a closed vector subspace of E . Let E be a norm ed space and M a subspace of E. W e denote by

:M

»E

(or sim ply

JM : M

the em bedding m ap. If M is closed in E , then E /M

►E)

becomes a norm ed space w ith respect

21

to th e q u o tie n t norm ||x(M )|| = inf{ ||x + m || : m

6

M } ,

w here x(M ) = x + M denotes the equivalence class containing x . T h e q u o tie n t m ap from E

o n to

E /M

is denoted by

E

or sim ply

or

E

Q . O f course,

is continuous,

m oreover it is open as shown by the following:

(2.6)

subspace of E and

(2.6.1)

hence

Let (E, ||-||)

Proposition.

: E ------- ►E /M

be a norm ed space, let

M

be a closed vector

th e q uotient m a p . T hen

° E /M = Q m ( °

e

)C

C UE /M ’

is an open o p e rato r.

Proof. It is easily seen th a t

c ^ e /M

s^nce

IIQm (x )II = inf{ ||x + u|| : u € M } < ||x||. T o prove th e converse, i.e.,

c Q jy f(°g ). let Q M (X)

inf{ l|x + u|| : u

6

u 0€ M such th a t

||Q m ( x )|| < ||x + uQ|| < 1 .

Now let x Q =

x

T im T hen x^ e 0 ^

Q m M = ^ M ^ Xo^ G ^ M ^ E ^ Finally, since ^M ^E ^

O ^ /M ' T h en

M } = ||Q M (x)|| < 1,

hence th e definition of infim um shows th a t there is a ( 2 .6 . 2 )

6

^M ^E ^ ’

(by (2.6.2)) is such th a t

( 2 -6 .2 )), hence

is continuous and hence th a t

0

^ /M C ^ M ^ E ^ * , we conclude th a t

22

Q m (Ue ) -

CW

“ ^ E /M “ UE /M •

If E is com plete (resp. separable) th en so

(2.c)

is

Q u o tien t spaces and pro d u ct spaces:

E /M as shown by

Let

(E ,|| *||)

th e following:

and

(E j , ||- || j )

(i = 1,2,- • • ,n) be norm ed spaces and M a closed vector subspace of E . (i) If

(E ,|| *||) is com plete (resp. separable), th en so

is the quo tien t space E /M

(equipped w ith th e q u o tie n t norm ). (ii) Consider th e cartesia n product x = (x 1 , • • • ,x ) 1 n

6

n II E-, and define, for any i= l

n II E-, th a t i= l 1

INI

= t

su p ||x II ; l< i< n 1 1

||x|| C-

= ( S j “ j llXjll; 2 ) 1/ 1-1 ' 1

2

IWI i = S i = i HxiHi .

T hen

|| • ||

, || • || t

and f

|| • ||

are norm s on t

II E- , and these norm topologies coincide i= l 1

n w ith th e product topology; m oreover, II E. is com plete (resp. separable) (for these norm s) i= l 1 if and only if each

( E J |- ||. ) is com plete (resp. separable). (U sually, the norm

||-||

is t

called th e product norm of || • ||. (i =

C (E .) = ( n E j , ||- II n 1 i= 1 1

1 , • • • ,n),

);

and we also w rite

^ (E .) = ( n E. II-IU ) and n 1 i= 1 1 t

4n( E j1) = (i =n 1 E.,11 • I£I 1 )-) 1 All norm s on a fin ite -d im en sio n a l vector space are equivalent as shown by th e following:

23

( 2 .7)T h eo rem .

(E ,|| • ||)

be an n—dim ensional norm ed

space,and let

} be a basis of E . T hen there exist num bers a and ft > 0 such th a t

{ e , • • • ,e 1

Let

n

(2.7.1)

o. °-

^ e i £ E , let

£i = — n— ~ — ii— (C .IA ^ V 2

(for all i =

1 ,*

• • ,n)

T hen [£j] e S, hence (2.7.3), (2.7.5) and (2.7.7) show th a t

_

—

n

1

T7

(E-J =1, IAJjl ) it th en follows from x = £

Ajei th a t

l|x|| > a(E._i | A j |2)2 .

IIV 11 \ „ II •

11 1=1

X

11

® ’

25

(2.8) Corollary, (a) Every finite—dim ensional norm ed space is c o m p le te . (b)

E very fin ite—dim ensional vector subspa.ce of a. norm ed space is closed.

Proof, (a) Let E basis of E. T hen

be an n—dim ensional norm ed space and let

1

n

be a

^ E under th e topological isom orphism T defined by

T (K il) =

As

{ e ,• • *,e }

(for all fo] e A

is com plete, it follows from I\ ^ E th a t E is com plete. (b)

G. T h en

Let G be a norm ed space and let M be an n—dim ensional vector subspace of M

is an n—dim ensional norm ed space under th e re la tiv e norm , hence

M

is

com plete by (a), thus M is closed in G (since the norm —topology is H ausdorff). As a consequence of ( 2 .8 ), we obtain the following in te restin g result.

(2.9) Corollary. basis of

E

If

E

isan infin ite-d im en sio n al

is u n c o u n ta b le . (A subset

B

of E

is a

B anach space, th e n H am el basis if

any Ham el

Bis linearly

in d ep en d en t and E = < B > .) Proof. Suppose th a t E has a countable H am el basis { e 1?e2, • • • }. F or any n > 1 , let M n = span ({ e1;- • • ,en }). T h en M n is closed (since dim M n = n) and E = U M n. By B aire's C ateg o ry theorem , th ere is som e

for some x e

containing a n o n -e m p ty open set; in p a rticu la r, N (x,r) C

and r > 0. C onsequently, rO E C M K

(since N (x,r) = x + rO ^ and

- x e M ), it then follows from K

E = U n O r th a t n

L

E

c Mv , K

and hence th a t dim E < k , which gives a contradiction. In order to give a characterization of finite—dim ensional norm ed spaces, we need the

26

following crucial lem m a:

(2.10) R ie s z 's L em m a. Let subspace of E

(2.10.1)

E

be a norm ed space and

(i.e., M = M / E ). F o r any e

6

M a pro p er, closed

vector

(0,1) there exists an x f e E such th a t

||x || = 1 and d is t( x f , M) = inf { ||x^ - m || : m e M} > e.

Proof.

It is clear th a t

||Q M(u)|| = dist(u,M )

(for any u

6

E). Now choose

z e E \M ; since Q M(z) / O (M ) and th e q u o tie n t norm || • || is a norm , it follows th a t

(2 . 1 0 .2 )

||Q m(z)|| - d ist(z,M ) = a >

0,

and hence from a < —— and th e definition of infim um th a t th ere is some y 0 e M such th a t

(2.10.3)

a < | | z - y 0| | < —

Let us define

z -yo

T hen ||x^|| = 1; m oreover, we have, for any u

l|xf ~ ull

= I ' P -yoll

6

M, th a t

UI1

=

W 1- yj -

||z - y o - I I 2 - y 0ll u||

=

||z - y 0ll

[ 2 ~ ( y o + I P - y o l l U )|

V= 6

(by (2.10.2) and (2.10.3)) since y 0 + ||z — y 0|| u e M, thus d ist(x M) > e .

an

27

R e m a rk .

In R iesz's lem m a, th e proper vector subspace

closed. F or instance, let

M

has to be necessarily

E = C[0,1] be equipped w ith th e sup—no rm

|| •

(for definition,

see (2.17) (f) below ) and let M be th e vector subspace of all polynom ials on

[0,1]. T hen

M = C[0,1] an d therefore th e result fails to work in th is case. Also, one cannot generally take in a B—space E such th a t

6

=

1

in R iesz's lem m a; in fact, th e existence of u

||u|| = 1 and dist(u,M ) > 1 is a c h ara c te riz a tio n of reflexivity of

E (see (3.d) th e next section or D iestel [1984, p .5—6 ]).

(2.d) F in ite —dim ensional B anach spaces: Let

E

be a finite—dim ensional norm ed

space an d M a proper, closed vector subspace of E. T hen th ere exists a

||u|| = 1

and

ueM

such th a t

dist (u,M ) = 1.

[Using R iesz's lem m a and Heine—B orel's theorem on finite—dim ensional space.]

(2.11)

T heorem (Riesz). A norm ed space E is finite—d im ensional if an d only if its

closed u n it ball U r is c o m p act. P ro o f. N ecessity. Follows from Heine—B orel's theorem . Sufficiency. W e assum e th a t

is com pact, b u t dim E =

oo,

and th e n show th a t

th is leads to a contradiction. Choose X j 6 E w ith HxJI =

1;

then M t = < { x j >

is an

1 —dim ensional

subspace

of E which is closed and proper, by Riesz's lem m a, th ere is som e x 2 e E such th a t

||x 2|| =

M 2 = < {x 1?x 2} >

1

and

||x 2 —x jl > — -

is a 2 —dim ensional proper closed subspace of E , by R iesz's lem m a, th ere

is an x 3 e E such th a t

||x3|| =

1

and

dist (x 3 ,M 2) > 1/2;

28

in p a rticu la r,

ll X 3 — X 2l! > ! / 2

and

l l x 3 — x ll l > V 2 -

C ontinue this process, we o b ta in a sequence {xn} in E such th a t ||x n|| =

and

1

||x n — x m|| >

1/2

whenever n / m.

O bviously {xn} cannot have any convergent subsequence. T his contradicts th e com pactness of fl

E

, hence dim E
C (called th e inner p ro d u c t) satisfying th e following (11)

[AjX^+A 9 X2 ,y] = A -Jxpy] + A9 [x9 ,y] for all x ^ x ^ ^ € H and A^ A£

(12)

[x,y] = [y , x] (th e bar denotes th e complex conjugate),

(13)

[x,x] > 0 for all x e H,

(14)

[x,x] = 0 if and only if x = 0.

6

C,

For a fixed y e H , (II) says th a t [*,y] is a linear functional on H, while for a fixed x e H, (II) and (12) in d ic a te th a t [x ,• ] is a conjugate—linear functional on H in the following sense

[ x ^ y j + z ^ y j j ] = £ 1 [x >y1 ]+ /‘2 [x >y2]F u rth erm o re, if we define

( 2 . 1 2 .a)

||x|| = [x,x]?

for all x e H,

29

th e n ||x|| > 0 , ||x|| =

0

if and only if x =

0

|| Ax|| — | A | ||x||

,and for all A G C and x G H.

T herefore it is n a tu ra l to ask w hether ||.|| is a norm . T he answ er is a ffirm ativ e as shown by th e follow ing result.

(2 .1 2 )

Lem m a. Let H be an inner product space. T hen

a)

I [x,y] | < ||x||||yj|

for all x,y

6

H

c o n se q u en tly , we have

(2 )

llx + y || < IMI + llyll

for all x,y

6

H.

P ro o f. If y = 0, th e n ( 1 ) is trivial. Therefore we assum e th a t y / 0. F o r any A G C,

0

< [x+Ay,x+Ay] = ||x || 2 +A[y,x]+A[x,y] + AX||y||2;

in p a rtic u la r, if A = —[x,y]||y|| 2, the last form ula is easily seen to becom e

w hich o b tain s ( 1 ). T o prove ( 2 ), we notice th a t

||x + y || 2 = [x+ y,x+ y] = ||x | | 2 + [x,y] + [y,x] + ||y | | 2 |x || 2 + 2Re[x,y] + ||y || 2 < ||x | | 2 + 2 1[x,y] | + ||y ||2. F o rm u la ( 2 ) then follows from ( 1 ).

F orm ula (1) is usually called C auchy—Schw arz's in e q u a lity .

30

T herefore every inner pro d u ct space is a norm ed space under the n o rm defined by ( 2 . 1 2 .a), w hich is called th e associated n o rm .

A H ilbert space is an in n er pro d u ct space which is com plete under th e associated norm ( 2 . 1 2 .a). (2.e) Some c h ara cte riz atio n s of inner product spaces: (a) Let H be an inner product space. Then: (i)

(P o larizatio n id e n tity ) For any x,y E H,

[x >y] = ^{llx + yll2 - l | x - y | | 2 + i||x + iy ||2 - i | | x - i y | | 2}. (ii)

(P arallelo g ram law ) F or any x,y E H,

llx + y || 2 + llx — yll 2 =

(b)

2 ||x | | 2

+

2

||y ||2.

A norm ed space (X ,|| *||) is a inner product space if and only if th e norm || • ||

satisfies th e p arallelogram law. [On H*H , th e function, defined by[x,y] = ^ { ||x + y | | 2 — ||x—y ||2 + i||x-hiy ||2 — i||x —iy ||2, is an inner product.]

Let H be an inner pro d u ct space. Two vectors x and y in H are said to be o rthogonal, denoted by x ± y, if [x,y] =

0.

A subset B of H is said to be orthogonal (resp.

o rth o n o rm al) if

x ± y for all x,y E B (resp. x ± y for all x,y e B and ||x ||= l) .

T he orthogonal com plem ent of B, denoted by B x , is defined by

B 1 = {u G H : [x,u]= 0 for all x E B}.

31

(2.f) Some properties of orthogonal com plem ents : Let H be an inner product space and B c H. (a)

(P ythagorean theorem ) If x ± y th en

l | x + y | | 2 = | |x ||2 + ||y | | 2.

(b)

B x is a closed vector subspace of H.

(c)

If B is an orthogonal set of n o n -z e ro vectors, th e n B is linearly

independent.

Let A be a n o n -e m p ty set. T hen th e fam ily 3(A ) of all n o n -e m p ty fin ite subsets of A is a d irected set under th e set inclusion. If [xi?A] is a fam ily in H , for any a E 3(A ) we w rite

x

hence

a

= £ • x •, iEa i ’

£ 5(A )} becom es a net in H which is called th e associated n et w ith [xi 5A]. If

{ x ^ ,a E 3(A )} is convergent under th e norm , then we w rite

E a X: =

A

1

1im

0 6 3

£:_

(A) *e a

£^Xj

to be its lim it, th a t is

X: .

'

T he following result is a generalization of th e P y th a g o rea n theorem . (2.13)

P roposition. Let H be a H ilbert space and [x^A] an orth o g o n al fam ily in H.

T h en th e following sta te m en ts are e q u iv a len t. (a) T h e n et {x^, a E 3(A)} associated w ith [xj,A] is c o n v erg en t. (b) EA || X i | | 2
(b) : In view of th e c ontinuity of the norm and th e P y th ag o rean theorem , it follows from th e existence of S ^ x. th a t

P Ax AI

= ||

||2

=

lim S a e 5 ( A) lim

X l|2

S.

=

lim p a e d (A ) *e a

||2 1

||x . | | 2 < oo,

f t£ 5 ( A ) *6 a

1

and hence th a t ||X ^x . | | 2 = £ ^ ||x . | | 2 < oo. (b)

4 (a) : F or any e > 0 th ere exists an a Q e 5(A ) such th a t

^ •i ec aJ ul x 1-ll" 2 ~

||2 < t e a o ||x. 11 i"

6

for all a > - ao .

If a ^ a ^ e 5(A ) are such th a t os > a Q (j = 1,2), then we have by the P y th ag o rean theorem th a t

llE* « V E* t t x ill2 = E l h l l 2 + ,PS , Hx iH2 i 2 sea \ a t ea \ a 1 2 2 i




1

}.

T h en it is easily seen th a t B = {0,1}

(N

(th e set of all m aps from IN into th e 2—point set { 0 ,1 })

a nd th a t

I k -J /ll^

1

for all f, 7/ e B

w ith

{ft],

hence B is no t countable since

card B = 2caic* ^ = 2^° = c

(th e power of th e continum ).

t

is not

36

It th en follows th a t separable, th u s

B

can n o t co ntain a countable dense subset; in other w ords, B is not

is no t separable. (Since it is easily shown th a t any subset of a separable

norm ed space m ust be separable.)

(b)

T he subsets, defined by

c = { £ = [fj E

: lim j

exists }

6

and

Co = { C = [CJ G

are vector subspaces of

: l i m ^ = 0 },

f°. E quipped w ith th e sup—norm

|| •

, c

and c 0 are separable

B anach spaces. F u rth e rm o re , for any n > 1 , let

e ( n > = [^•n )]j >1 and

e = ( 1 , 1 ,* * *),

w here $ n)

T hen

i = |

1

if j = n

1

0

if j / n .

the countable set { e ( n ) : n > l } u { e }

{ e ( n) :n >

1

is a Schauder

basis

for c ,while

} is a S chauder basis for c 0 .

Remark. T he space

c contains c 0

as a subspace,

and

thus

c ~

c0

©< {e } >

E very convergent sequence [£n] E c can be represented in the form

c 0 has codim ension

one,

37

[ £ J = K 1 + Ae with A = i?m fj and [%] = [£„ - A] J

(As lim j

J

= A, it follows th a t lim sup I A- — AI = n j>n J

C -

=

A

e ( k ) - E>

and hence th a t

i - A)e < j , - H »

l i ( A, * * * , A, Cn+1, . .

sup

0,

-

Ael l^

| 6 - A |

►O

(as n

►oo).)

j >n+l (c) T he vector space, defined by

C = { e = [€nl e

=Sn |

|

1 ),

O n th e o th e r hand, for any e > 0, p ^ > n

l\l

> 1 and [rjJ e ^Po, th e re is som e N >

1

such th a t

1 /p

°)

° < 6

(for a11 n - N ) ‘

F or all p w ith p > p^ , we obtain from (e .l) th a t

p ( s 00, \ n , \ k =1 k

1 /p

m P 1/P 1) P1 d m (t)) P2 (b —a)

f b J a | x (t) |P* d m (t)

P2

rh JB i P2 ~ Pi = (Ja ( |x ( t ) |P 2 d m (t)) P 2 ( b - a ) P2 , thus

P2 - Pi !lx llp,

/•h

n,

= ( / a b | x ( t ) | P 1 d m (t ))

^P ,

1 < ||x||p (b - a)

_ J_____ 1_ = ||x||

T his proves (i.l) and L

2 [a,b]

P2

(b -a)

c L ^ b ].

T o prove (i .2 ), let x( •) e L°°[a,b] and

A = 11x11 = ess su p | x M | . 00 [a,b] T hen

|x ( * )| 00

j

= sup ||x|| P

P >1

(for all p > 1), so th a t

< A. P

O n th e o th e r hand, for any e > 0, there exists a Lebesgue m easu rab le set B 0 in [a,b] such th a t m (B 0) =

0

and

sup |x ( t ) | > A — e. t e [ a , b ]\B 0

Now th e set B = [a,b ]\B 0 is m easurable w ith 0 < m (B ) < b — a such th a t

1

| | x | | p > ( / B |K (t) lP d m (t)) P

1

> (A — e) (m (B )) p ,

so th a t

i l m ||x|| P~i 00

> A - e. 1

T hus

l i m ||x||

= A = ||x||

P ~ 100

By a sim ilar argum ent give in the proof of

( a ) --------- (e) of (2.17), one can verify

th e follow ing m ore general B anach (sequence) spaces.

( 2 .g) T he spaces / P (A) (I): Let A be a n o n -e m p ty index set. T h en th e collection of all n o n -e m p ty finite subsets of A, denoted by ^ A ) , is a directed set ordered by th e set inclusion. E lem ents in 3{K) will be denoted by n, /?,

7

etc. A fam ily [Ai?A] of num bers is

said to be sum m able if th e n et { E - ^ ^ A .^ A ) } converges. T h e uniquely d eterm in ed lim it A is called th e sum of [Ai?A], and denoted by

44

A=

(i)

A

fam ily

[A-,A]

or

of

A = S. A.

num bers

is

absolutely sum m able in th e sense th a t th e fam ily sum m able, and this is th e case (ii) If [A-,A]

if and only if

sum m able

if

[ | A-1 ,A] su p

S.

a e ^ A ) l€a 1

and

only

of positive I A-1
0 such th a t

|| T x || < A || x ||

(for all x e E).

T h e following result shows th a t L(E, F) is a norm ed space un d er th e o p erato r norm :

(3.1)

T h eo re m . Let E and F be norm ed spaces over K . T h en L (E , F ) becomes

a n orm ed space over K . under the operator norm ||-|| , defined by

|| T || = sup { || T x || : x e U E }

47

(for all T 6 L (E, F )).

48

i M oreover, if F is co m p lete, th en so is

L IE, F). In p a rtic u la r, th e topological dual E

of

E is alw ays a B anach space (called th e B anach dual of E). T he norm —topology on

L (E , F ), induced by the o perator norm , is called th e

uniform norm —topology.

R e m a rk . T he o p e rato r norm can be represented by

(3.1.1)

|| T ||

= inf { a > 0 : || T x || < a || x || for all x e E } = sup { ||T x || : x 6 0 E }

= sup { || T x || : || x || = 1 } = sup {

l^jjH

: 0 # x € E } (for all T € L (E, F )).

T he proof is rou tin e, hence will be om itted.

(3.b) Isom orphic em b ed d in g : Let E and F be norm ed spaces and T

g

L (E, F ). W e

say th a t T is isom orphic em bedding if T has a bounded inverse T * : T (E )— >E. It is not hard to show th a t T is isom orphic em bedding if and only if

inf { || T x || : || x || = 1 } > 0 (com pared w ith (3.1.1)) or e quivalently, there exists an r > 0 such th a t || T x || > r || x || (for all x G E). (3.c) L inear m appings betw een finite—dim ensional B anach spaces: Let E and F be finite dim ensional B anach spaces. T hen any linear m ap Consequently, i * dim E = dim E = dim E < oo.

T :E

►F

is continuous.

49

(3.2) Theorem (Induced m apping theorem ). Let E, F and G be norm ed spaces over K . G iven th e following operators E —

>F

S| G

Suppose th a t S is suriective and th a t (3.2.1)

|| Tx || < A || Sx || (for all

x e

E)

for som e A > 0. T hen there exists a unique R € L (G ,F ) such th a t (3.2.2)

md

T = RS

|| R || < A.

Proof. For any y e G, the surjectivity of S ensures th a t th ere is an x e E such th a t y = Sx , hence we define R on G by setting (3.2.3)

Ry = Tx.

(R em em ber th a t x e E

is such th a t y = Sx.) (3.2.1) im plies th a t R is w ell-defined since

K er S c K er T . [Let y = 0. T hen 0 = y = Sx im plies th a t x e K er S c K er T , hence 0 = T x = Ry.] C learly R is linear and satisfies RS = T (by (3.2.3)). T o prove th e c o n tin u ity of R, let y e G and let x e E be such th a t y = Sx . T hen (3.2.1) shows th a t

II Ry II = II T x II < A II Sx II = A II y ||, so th a t || R || < A. F in ally , the uniqueness follows from th e surjectiv ity of S. F or a dual result of (3.2), see Ex. 3—17. (3.3) Theorem (H ah n -B an ach ).

Let

E

be a norm ed space and

M

a vector

subspace of E. F or any continuous linear functional f0 on M, th ere is an f e E ' w ith f(m ) = f0(m ) for all m e M (called an extension of f0) such th a t

50

II f II = II fo Hm (= su P { l fo(m )I : m € M , ||m || < 1 }).

P roof. For any x G E , let

(3.3.1)

p(x) = || f0 ||M || x ||.

T hen p is a norm on E such th a t |fo (m )| < ||f0|lM ||m || = p(m )

(for all m e M).

* By th e H ahn—B anach extension theorem , th ere exists an f e E extending fo such th a t | f(x) I < p(x)

(for all x £ E). I , and hence th a t f G E . On th e o th er h and, if

It th en follows from (3.3.1) th a t || f || < || fo || M m G M , then

If0(m )I = |f ( m ) | < ||f|| ||m ||, hence ||f0||M < ||f||; consequently, ||f|| = ||f0||M -

(3.4)

C orollary.

Let

E

be a norm ed space. For any

0 / xo G E, th ere exists an

i f GE

such th a t

(3.4.1)

|| f || = 1

and

f(x0) = || x0 ||.

C onsequently. (3.4.2)

|| x ||

= sup { | g(x) | : g € E , || g || = 1 } = sup { | g(x) | : g € E , || g || < 1 }.

P roof. Let M = { Axq : A g K} and define

51

fo (A xo)

= A || xo ||

(for all A 6 K ).

T h en M is a subspace of E and fo is a continuous linear fun ctio n al on M w ith

|| fo || = 1

hence (3.4.1) follows from (3.3). T o prove (3.4.2), we first notice th a t it is triv ia l for x = 0 , hence we assum e th a t x / 0. As

|h ( x ) | < ||h|| ||x||

(for all h e E*),

it follows th a t sup { |g ( x ) | : g £ E , ||g|| = 1 } < sup { |h ( x ) | : h e E , ||h|| < 1 } < ||x||. O n th e o th e r hand, for any fe E

O^ x e E ,

the first p a rt of (3.4) shows th a t th e re exists an

such th a t ||f|| = 1 and f(x) = ||x|| ,

so th a t ||x|| = f(x) < sup { |g ( x ) | : g 6 E , ||g|| = 1 } . T h u s (3.4.2) holds.

i Let

E

be a norm ed space and let

E

be th e B anach dual of E

. T h en

i E is a

B anach space un d er th e norm

||f|| — sup { | f(x) | : x e U E }

(for all f e

e

').

i

T h e B anach dual of E , denoted by E" , is called th e B anach bid u al of E. W e w rite th e e v alu ate m ap from E into E" , defined by

for

52

< f, K £ x > = < x , f> = f(x)

(for all f €

e

’),

th en (3.4.2) shows th a t

I|Ke

x

|| =

||x||.

A B anach space E is said to be reflexive if

is suriective.

F rom (3.c), any finite—dim ensional B anach space m ust be reflexive. It can be show n (see (a) and (b) of (3.9) below) th a t th ere exists an infinite-dim ensional B anach space w hich is not reflexive. As an ap plication of (3.4),we verify the following result which is very useful for studying approxim ation theory.

(3.5)

T heorem . Let E be a norm ed space, let N be a vector subspace of E , and

let uo G E be such th a t

p = d ist (uo, N) = inf { ||uo — w|| : w e N } > 0.

(3.5.1)

T hen th ere exists an f G E

(0 (ii) (hi)

such th a t

llfll - 1; f(u0) = dist (u 0,N); N c f ^ O ).

P ro o f. It is easily seen th a t

d ist (x,N ) = dist (x,N )

(for a n y x G E);

and th a t || Qpq-(x) || = d ist (x,N )

(for all x G E),

53

w here

: E

►E / ^ is the q uotient m ap. It th e n follows from (3.5.1) th a t Q ^ ( u 0) /

0(N ) (since || Q ^ ( u 0) || = dist (u0, N) = p > 0), and hence from (3.4) th a t th ere exists an i f e (E /j^ ) such th a t || f || = 1 and f( Qjq-( u 0 ) ) = II Q ^ K ) || = p.

I__________________________ _ T hus f = fo Q j^ e E

has th e required properties (since N c N = K er

and O j ^ _

= Q ^ ( ° e ))-

(3.d) Let E be a Banach space. (i) If

f e E is such th a t ||f|| = 1, then dist (x, f_1(o)) = |f(x ) | (for any x e E).

(ii) E is reflexive if and only if for any p roper closed vector subspace M of E th ere exists an 0 / u 6 E such th a t dist (u, M) = ||u||. [The necessity follows from (3.5) and (3.3), while th e sufficiency follows from (i) and J a m e s ' th eo rem (see Jam es [1964]), which states th a t a B anach space G is reflexive if and only if i any f e G a tta in s its suprem um on U q .]

i

(3.e) S e p a rab ility : m u st be separable; separable. [Let

A norm ed space E for w hich its B anach dual

E isseparable

th ere exists a separable B anach

space w hose B anach dual is not i { fn : n > 1 } be a countable dense subset of E and let x n e U g be such

th a t |f ( x n ) | > i | | f j | . T hen

M = {

rjaj : aj6 { x n : n > 1 } and

in

are ra tio n al }

is cou n tab le, and use (3.5) to show th a t M is dense in E.] t Let

E

and

F

be norm ed spaces and

T e L (E , F ). W e denote by T

th e

54

i * restrictio n on F of th e algebraic a d jo in t T of T; th a t is,

< x , T y '> = < x , T y '> = < T x , y '>

(for all x G E and y ' G F ') .

t i It th en follows th a t T y ' G E is such th a t

IITy'll < IIT|| ||y'|l i and hence th a t

T G L (F , E ) and ||T || < ||T || (it can be shown th a t ||T || = ||T ||). T is i called th e dual operato r of T . T h e dual o p e rato r of T , denoted by T" , is referred to as

the bidual o p erato r of T. C learly, T " : E "

►F" is an operato r such th a t

M oreover, it is easily seen th a t I , - (I = 0 (for all b 6 B) }, t

is called th e an n ih ilato r of B. D ually, if D c E Dt = (

X

then th e annih ilato r of D is defined by

6 E : < x , d '> = 0 (for all d ' 6 D) }.

(3.f) D ual operators and a n n ih ila to rs: Let E, F be norm ed spaces, let T G L(E, F), let B c E and D c E . (i) I|t ’|| = ||T || (ii) B

-L

' XT ( resp. DT ) is a closed vector subspace of E (resp. E) ;m oreover, (B ) is

the sm allest closed vector subspace of E containing B. [Using (3.5).] As an application of (3.3). W e verify th e following:

(3.6)

T heorem . Let

(E, ||.||)

be a norm ed space and

M a closed vector subspace

I , of E. (T h e q u o tie n t norm on E /M (resp. E /M ) is still denoted by ||.||). i (a) T h e H ahn—B anach extension theorem (see (3.3)) ex ten d s each m ' e M

to an

i x ' E E , hence we define

ip(m') = x ' -f M -1

(3.6.1) i i T hen M e E / M

,

under th e m etric isom orphism (i.e., iso m e trv ) ib .

►E /M be th e q u otient m a p , and let Q ^ : (E /M ) -------►E , \ be its dual m a p . T hen (E /M ) = M under th e m etric isom orphism Q (b) Let

: E

t

P ro o f, (a) If x '

i i . and x j , in E , are extensions of m ' e M , th en x ' — x j e M ,

hence x ' + M -1 = x j + M x , thus ip is w ell-defined. As th e re stric tio n of every x ' e E to i i . M is continuous on M, it follows th a t ip : M >E /M is surjective. It is easy to check ip is linear, it rem ains to show th a t ip is a m etric isom orphism . T o do this, let i i m ' e M . B y (3.3), m ' has an extension x ' 6 E w ith th a t

(3.6.2)

Hx'll = l |m '||M = sup { |< m , m '> | : m e U M }.

T h u s x ' + IVC is the set consisting of all (continuous and linear) extensions of m '. C learly if u ' G E ' is an extension of

(3.6.3)

m ', then ||m '||j ^ < | | | | , hence

llm ' | | M < inf { ||x ' + u '|| : u ' e Mx } = || x ' + M ± ||

(by th e definition of quo tien t norm and x ' + M ). On th e o th er hand,

||x ' + M || = inf { ||x ' + u '|| : u ' G M W e conclude from (3.6.2) th a t

} < ||x '||.

56

and hence from (3.6.3) th a t

llm ' IIm = Hx ' + M ±H =

(b) T he c o n tin u ity of W e first claim th a t

im plies Q ^ 6 L ((E /M ) , E ) .

is an isom etry (in to ), th a t is

(3.6.4)

IIQM (y ')ll = lly'll

(for all y ' € (E /M )').

In fact, we first notice from (2.6) th a t

° E /M = Q M ( 0

e

) C Q M (U e ) C U E /M ■

I Now for any y ' G (E /M ) , we have

I I QM ( y ' ) l l

= su p

|< x , Q M ( y / ) > l = s u p

= su p xe0

|< X , y ' > | = ||y ' E /M

which proves our assertion . W e com plete th e proof by show ing th a t

(3.6.5)

Q m ( ( E /M ) ’) = M \

Indeed, let u ' e M

i

.T hen we have th e following operators

E /M such th a t

is surjective and

l< Q M x, y '> |

57

Ker

= M c Ker u '.

Hence, th e re exists a linear functional y '

on E /M such th a t

y ' 0 Qm = u ' •

As °

e

/ m = cW llu 'll

°

e

) ’ we obtain

= sup{ | < x ,u '> | : x e 0 E } = sup { | | : x 6 O e } = ||y '|| , 1 hence y ' e (E /M )

is such th a t u ' = y ' o

I = Q ^ j( y ') , th u s (3.6.5) holds.

C om bining (3.6.4) and (3.6.5), th e result follows.

It is know n (see (3.c)) th a t every finite—dim ensional B anach space is reflexive, and th a t th ere are infinite dim ensional B anach spaces w hich are, in general, n o t reflexive. H ow ever, every point in E" is related to a point in E by m eans of finite dim ensional i subspace of E as shown by the following im p o rta n t and useful result:

(3.7) and let

N

T heorem (H e lle y 's selection theorem ). Let E

be a finite—dim ensional vector subspa.ce of E

be a

.F or any

norm ed space,let eE"

e > 0th ere exists

uo E E such th a t

lluoll < I l l 'l l + e and

K ^uo =

on N.

P ro o f. T he an nihilator of N (in E) i.e.,

M = NT = { x e E : < x , n '> = 0 (for all n '

e N)

}

an

58

is a closed vector subspace of E such th a t

M ±= ( N t )-l = N I (since dim N < oo and surely N is closed in E ). Let

(3.7.1)

- N x = { x" = 0 (for all n ' e N ) }

(i.e., th e a n n ih ilato r of N in E "). T h en (3.6) shows th a t

(3.7.2)

( E /M )'

e

M '

N and N =

e

so th a t

(3.7.3)

( E /M ) 11

e

N

e

E " /M j" l .

It th en follows th a t th e following diagram com m utes Kr

E (3.7.4)

Q

E" QM

M E /M ----------KE/M I

[ For any x e E

, since

N

= E " /j^ ± ±

< n ', (Q j^j.± o K ^ )x > = < n ', (3.7.1) < n ',

(3.7.5)

o Q ^ ) x > (for all n 'e N ) . Indeed, we have from

th a t Q ^ ± ± ( K g x )>

< n ', (K g^jyj o Q ^ ) x > (3.7.4)

(by (3.7.2)), it is required to show th a t

=

< n ', =

KgX-t-M±J'>

=

< n ',

K ^x>

< Q j^ x , n '> = < x + M, n '> = < x , n '>

holds.] M oreover, we claim th a t

E /M _ E " / ^ ± ±

In fact,since dim N
0 , th ere exists

a

uo G E such th a t

||uo|| < A + c and = lq (i = 1,* •- ,n).

Proof. W e em ploy (3.7) to verify this theorem . Clearly

N = < { x ;,- • - ,x '} >

is a finite—dim ensional vector subspace of E . Let us define p on N by

(2)

=

< x ' i? K e u 0 >

=

1).

i =l

T hen (c.5) becom es = f of l i m ( ) + E “ Ci/(i4i n

(for all ( = [(„] € c).

i =l

W e com plete th e proof by show ing (see(c.3)) th a t

(c.7)

Indeed, let us define

/ ( = [ / i j 6 < 1 and

||/i||, < ||f||.

67

and M -

if Mi / 0

1

if /q = 0

T h en th e sequence, defined by

belongs to c and satisfies ||x ( n ) ||

oo

=1

and l i m x n = eo (w here x n+i = eo) (for all i > 1). n

By (c.5) an d (c.6), we have < x ( n ) , f> =

00 '"i+1 p xj u n + £ . =1 '(i i+1 x j + £ i=n+1

~ foA + Si =1 l/h+ll + ^°=n+1 ^ i+ l€0 hence II > | < x ( n ) , f > | > |fo/*l +

l/V ll | -

^ L ettin g n -

►oo, we o b tain £°° |/ q | < ||f|| , which proves our assertion (c.7). i =l

(d) For fe(fP)'

(d .l)

|£01 = 1 ) -

1 < p
= £ “ Cj m

w here t] = [j?i] 6 A

and

(for all C = [Cj 6 ^)>

= 1); m ore precisely, every

68

||f|| = M q = ( E “

(d.2)

h / i |q ) q

•

Proof. T he arg u m en t is very sim ilar to th a t used in the proof of E xam ples (a) and (b), hence will be o m itted . For the B anach duals of L ^[a,b], we m ention the following results, whose proofs can be found in th e book of R oyden [1968, P . 117—118, P . 121—123].

(e) For 1 < p
= f

(e .l)

n

x (t)y (t) d m (t)

(for all x ( •) G L^[a,b]),

a where y( - ) 6 L^[a,b] is such th a t -

b

( f J y ^ ) I qd m (t) )q

if p

t 1

(e.2) e s s sup | y (-) I [a ,b ]

if P = 1 ■

T he proof of th e following im p o rta n t result, due to Riesz, can be found in the book of R udin [1966, p .40 and p .130].

(f) Let Q be a com pact

H ausdorff space and let M (ft) be the set of all com plex

regular Borel m easures (called com plex R adon m easures) on

Q . T hen

M (H)

becom es a

B anach space under the norm

(f.l)

||//|| = |/ / |( f i ) = sup { E .nJ /z ( B i) | : Bi are Borel sets w ith B i n Bj = ,A ] € ^(A ) (for any u ' G E ); (B) sum m able if th e net { Ej (in E

x p a G ,A ] G A(A) (for any u G E).

In p a rticu la r, if A = IN and if a sequence

{xn} in

sum m able, absolutely sum m able), then the form al series

E

is w eakly sum m able (resp.

E xn

is said to be weakly

70

unconditionally convergent (resp. u n conditionally convergent, absolutely c onvergent). If a i * sequence { x '} in E is w eak sum m able, then th e form al series X x ' is said to be * weak unconditionally convergent.

(i) If [xpA]

is a w eakly sum m able fam ily in

E

and

[Ai? A] e c0(A) , th e n th e

fam ily [Aixi, A] in E is sum m able ([see P ie tsch [1972, p .26]). (ii) E very absolutely sum m able fam ily in

E is sum m able (by th e com pleteness of

E (com pared w ith (2.b) (iii)). H ow ever,as th e fam ous theorem of D voretzky and Rogers says, th e converse im plication holds only for fin ite-dim ensional Banach spaces (see D ay [1973], G rothendieck [1956] or De G ra n d e —De K im pe [1977]). (iii) E very sum m able fam ily in a B anach space E contains a t m ost countably m any non—zero term s

(com pare w ith (2.e) (ii)) .

(iv) T he com pleteness of E ensures th a t the form al series X x n (w here x n 6 E) is n

unconditionally convergent if and only if it is bounded—m ultiplier convergent in th e sense th a t for any [f n] e Z50, lim E

exists in E (see Day [1973,p .78] or L indenstrass

/T iz afriri [1977, p .15]). (vi) A sequence [x ^ ] in E is absolutely sum m able if and only if S ||xn|| ^ oo. [By n

using (2.e) (i).] (vii) If [xj,A] is a w eakly sum m able fam ily in E then th e set { Ejg a x i : a 6 ^(A ) } is bounded; consequently,

|| [xi,A] ||f = sup { XA |< X i , u '> | : u ' e U E, } < oo. * Sim ilarly, if [x-, A] is a weak

i sum m able fam ily in E ,then

sup { XA |< u , x i ' > | : u e U E } < oo.

(3.j) O p erato rs in (or fro m ) concrete B anach spaces (I): Let E and F be B anach spaces. If T is a linear m ap from E in to e ith e r #(A ) or co(A) or ^°(A), th en we denote

71

th e i—th coo rd in ate of Tx by (Tx)i, hence

(Tx)i = in p a r tic u la r , if

and

Tx = [(Tx)i,A] (x e E);

Tx G #(A) (o r Co(A)), th e n Tx = EA(Tx)ie(i) in £(A )

(or c0(A)). *

(i) A linear m ap T : E

►£(A ) is continuous if and only if th e re exists a weak

sum m able fam ily [fi,A] in E such th a t

T x = [< x,fi>,A ] = E ^ < x ,f j> e ( 1]

(in £ (A ))

(for all x G E).

i In th is case, we have fi = T e ( 1} ( i e A ) and l|T|| = sup { SA |< x , f i > | : x e U E }.

(ii) A linear m ap T : E ►c0(A) is continuous if and only if th ere exists a * i bounded, w eak null fam ily [gi,A] in E (i.e., for any x G E, [, A] G c0(A)) such th a t

T x = [< x, g i > ,A] = ^ < x , g i > e ( 1}

0 n co(A))

(for all x G E ).

In th is case, we have gi = T e ( 1} ( i e A ) and

||T|| = su p ||gi||. (Dasor [1976,p.l06] for th e case A = IN.) ieA (iii) A linear m ap

T :E ► i bounded fam ily [hi, A] in E such th a t

Tx = [ ,A] i In th e case, we have hi = T e ( l) (i e A) and

is continuous if a n d only if th ere exists a

(for all x G E).

72

||T || = su p ||hi|| ieA S : il(A )

( i) ' A linear m ap

(P ietsch [1980, p .337])

►F

is continuous if and only if th ere exists a

bounded fam ily [yj, A] in F such th a t

S([Ci,A]) = EA Cjyi (for all [(;, A] e * (A )).

In this case, we have yi = S e( l)

(i 6 A) and

||S|| = su p ||y i||. iGA ( ii) '

A linear m ap

(K othe[1969, p.280],Pietsch[1980,p.34].)

S : C o ( A ) ----- ►F

is continuous if and only if th ere exists a

weakly sum m able fam ily [yi, A] in F such th a t

S ([Ai, A]) - SAAiyi

In this case, we have yi = S e ( 1} ( i G A )

(for all [Ap A] G c0(A)).

and

IISII = sup {EA |< y ; , g > |

:g e Up, } (= ||[yi,A ]||f ).

(G rothendieck[1956], D azor [1976,pp. 106—107] for the case A = IN.) (iii)' F or any u n conditionally convergent series E y n in F , the m ap, defined by n

S( W ) =

is a continuous linear m ap from

/'ny„ (for

all [na] e ( ° ) ,

into F w ith y n = Se ,n)

(n > 1) such th a t

||S|| < s u p { £ n” | < y n, g > | : g £ U p, }.

73

As a consequence of (3.j) and

H ahn—B a n a c h 's extension th eo rem (see (3.4)), one

can verify th a t f°(A ) (resp. £(A )) has the m etric extension (resp. m etric lifting) property, as defined in §6 (see(6.f)).

i (3.k) L inear functionals (II): Let

E be a real norm ed space, let f, g 6 E

be such

th a t ||f|| = ||g|| = 1 and e > 0 . If

|g ( x)l < - | - | l x ll

(for all x g f-‘(0)),

th en one can use (3.3) to conclude th a t either | | f - g | | < e or ||f + g|| < e. [The preceding result is useful for proving

Bishop—P h e lp s '

su p p o rtin g

theorem , see

B ish o p /P h e lp s [1963].]

(3.m ) R iesz's representation theorem for continuous linear functionals on H ilbert spaces. Let H be a H ilbert space w ith th e topological dual H '. F o r any ^ e H r, th ere exists a unique y E H such th a t

ip(x) = [x,y]

(for all x E H).

M oreover, th e m ap (j) : y i— > ^ is a conjugate—linear iso m etry from H o n to H' in th e following sense:

0 such th a t (1)

th en

||T x|| < Ax

(for all T e St).

St is bounded w ith respect to the operator norm (called uniform ly b o u n d e d ), th a t is,

th ere is an A > 0 such th a t sup { ||T ||: T G S t } < A .

P ro o f. For any n a tu ra l num ber n > 1 , let C n

T hen each

C

n

is closed in

E

= { x G E : s u p ||T x|| < n }. Test

(by the continuity) and

B a ire 's category theorem , there is some k > 1 such th a t

00 E = U C n =l u

co n tain s a n o n -e m p ty open

set G. C onsequently, th ere is some open ball N(z, e) = { z G E : ||x —z|| < t } w ith N (z, e) C C^,

75

(by (1)), hence by

76

th a t is, ||Tx|| < k As N (z, c) = z + N(0,

(for all T e ^

and x € N(z, e)).

e), for any u € N (0,e), we have ||Tu||

< |[ T(u + z) - T z || < || T(u + z) || + ||Tz|| < 2k

(for all T e # ) ,

hence

s u p ||T || < TeM

• (

1 (4.2) C orollary. Let E be a norm ed space, let B C E and D c E . i Bis bounded if and only if for anv u ' € E ,

(a)

su p | < b ,u ' > | < beB

oo.

i (b)

Assum e th a t

E

is co m p lete. T hen

D is bounded in the B anach dual

E

if

and only if for anv x e E, su p |

0

(or

T (O ^ )) is a

such th a t

0 —neighbourhood

78

r U p C T (U e ) (or r U p C T ( 0 £ )) .

(4.5) L em m a.

Let

E

an d

F

be norm ed spaces over

K and let

T : E --------->F

be lin e a r. (a) If T (E )

is of th e second category (in p a rtic u la r, T (E )

is co m plete), then

T is alm ost open. (b) Suppose th a t

T

is alm ost o p e n . If

E

is com plete and

T

is contin u o u s,

th en T is open (and a fo rtio ri, o n to ) hence F m ust be com plete.

Proof,

(a) As

T (E ) =

U n T (U ^ ) (since n> l ^

second category, it follows th a t one of th e sets

E =

U nU r ) n> l ^

n T (U p )

and

T (E )

is of the

has an interior point, and hence

th a t T (U p ) also has an in te rio r p o in t, y say; consequently,

0

= y + (—y) is an interio r point

of T (U p ) + T (U p ) (since T (U p ) is circled ). N ote th a t

T (U e ) + T (U e ) C T (U e ) + T (U e ) =

it

then follows th a t

0is an in te rio r point of

2 T (U e

2 T (U p )

6

2 T (U e

) ;

, and hence th a t

interior point of T (U p ); in o th e r w ords, th ere exists an r >

rU F = { y

) =

0

0m ust be an

such th a t

F : ||y|| < r } C T (U £ ) ,

th a t is, T is alm ost open.

(b) we claim th a t

T here exists an r > 0 such th a t rU p c T (U p ) (by th e alm ost openness of T );

4

t u f c t ( u e ),

so t h a t th e openness of T follows. T o this end, we first notice th a t

(2)

- A u F c T r T ( u E) = 2

t

(for all n > 1 ).

( - L . UE )

2

2

Now for a n y y E r U p , th e r e e x is ts a n x i E U p su c h t h a t

H y -T x J I < — | —

(since

rU p c T ( U p ) ) . As y - T xi E

X2 E —

U p , it follows from

(2

U p such th a t

lly-Tx1- T x

2

||
0.

are only

100

(6.3)

D efintion. Let E and F be norm ed spaces over K and T e L (E ,F ). W e say

th a t T is a (a)

topological injection for isom orphic em bedding), denoted by T : E >

>F, if

j(T) > 0; (b)

m etric injection if j(T ) = 1 =

||T || ;

(c)

topological su rie ctio n , denoted by T : E

(d)

m etric suriection if q(T ) = 1 =

F, if q(T ) > 0 ;

||T ||.

R e m a rk . Suppose th a t T e L (E ,F ). T hen T

is a topological surjection if an d only

if it is open; T is an isom orphism if and only if it is a topological injection and topological surjection; and

T is a m etric isom orphism (i.e. isom etry) if and only if it is a m etric

injection and a m etric surjection.

(6.4)

P ro p o sitio n .

Let

E

and

F

be B anach spaces an d

T hen th e following sta te m e n ts are e q u iv a le n t: (a)

T is a topological in je c tio n .

(b)

T is one—one and has a closed range T (E ).

(c)

T adm its a factorization

E

w here T 0 : E

T

F

T (E ) is an isom orphism and

In this case, we have

(6.4.1)

j(T ) = IITo’H 1.

is th e em bedding m a p .

T e L (E ,F ).

101

P ro o f, (a) => (b):

As j(T ) > 0 , for any

a

0 < a < j(T ), th e re exists an

w ith

r > 0 such th a t r > a

(1)

and

r||x || < ||T x||

(for all x e E).

It th e n follows th a t T is o n e -o n e [since T x = 0 $ x = 0 ]. T o prove th e closedness of T (E ), let

Txn

►y

in F. T h en

{xn} is a C auchy

sequence in E( by (1)), hence converges to some x e E (by th e com pleteness of E ), thus lim T x n = y = T x e T (E ). n

(b) => (c): By B a n a c h 's open m apping theorem , th e sta te m e n t (b) ensures th a t the m ap T 0, defined by T 0x = T x

(for all x e E ),

is an isom orphism from E onto the B anach space T (E ). T hus th e im p licatio n follows. (c) => (a): Since T 0 has a bounded inverse To : T ( E )

IWI = l|T il(Tx)|| < HTo‘ 11 ||Tx||

►E , it follows th a t

(for all x 6 E),

and hence th a t

(2)

j(T ) > U To'lf1 > 0 .

T herefore T is a topological injection. Finally, we assum e th a t T is a topological injection. T h en j(T ) > HT^H 1 (by (2)). In order to verify (6.4.1), it suffices to show th a t for any r > 0 w ith r||x || < ||T x || (x e E ), we have

i-IITo1!! < i.

102

Indeed, let r > 0 be such a num ber. T hen

r||To*||

= r sup{ ||T51(T x)|| : ||Tx|| < 1 } = sup{ r||x || : ||T x || < 1 } < sup{ ||T x|| : ||T x|| < 1 } < 1,

which obtains our assertion.

(6.5)

P roposition. Let E and F be B anach spaces and T e L (E ,F ). T h en th e following

sta te m en ts are e q u iv a len t: (a) T is a topological su rje c tio n . (b) T is o n to . (c) T adm its a facto rizatio n

Q \

T0 E /K e r T

w here T 0 : E /K e r T

►F is an isom orphism and Q is the q uotient m a p .

In this case, we have

(6.5.1)

q (T )=

||T 0' | f '.

Proof, (a) ^ (b): T riv ial (O pen m appings m ust be onto). (b) $ (c): Follows from B a n a c h 's open m apping theorem and the fact th a t

T 0 is

continuous and open (since T is o nto and E /K e r T and F are B anach spaces). (c) $ (a): As T 0 and Q are open, it follows from T = T 0Q th a t T is open. Finally, to prove (6.5.1) , we first notice th a t if r > 0 is such th a t th en

rU p C T ( U p ),

103

HTo'MI

= sup{ HTqVII : y G U F }

< sup{ IITqH ^ T x )!! : x G U E }

= r _1sup{ ||Q x|| : x G U p } < r ' 1,

hence

t

< HT^1)!-1,

and consequently, q(T ) < HT^H"1.

C onversely, let

r = HT^H'1. W e

claim th a t rO p C T (U e ),

so th a t rU p = rO p C T (U p ); consequently, r < q(T ). In fact, for T = T 0Q a n d

T0

any

y G O p, there is an x G E w ith

is an

isom orphism , it follows from

ry = T x

ry = T x

[since T is onto]. As

th a t

r T ^ y = T j^T x = Qx,

and hence th a t

IIQxIl = r||T o‘y|| < r ||T ; ‘|| ||y|| = ||y|| < 1.

N otice th a t Q (O p ) = O gyj^er ^ • T here exists a n u G O p such th a t

Q x = Q u ; it

th e n follows from T = T qo Q and ry = T x th a t

ry = T x = (T oQ )x = T o(Q u) = T u e T (O e ) C T (U E ),

w hich o b tain s our assertion.

F or c rite ria of topological injections (resp. topological surjections) betw een norm ed spaces, we q u o te th e following two results:

(6.b)

Topological injections and subspaces (see Ju n e k [1983]): L et

norm ed spaces and T G L (E, F). T hen th e following sta te m e n ts a re e quivalent (i)

T is a topological injection.

E

and

F be

104

(ii)

T is one—one and relativ ely open.

(iii)

(U niversal p ro p e rty of topological injections and subspaces) T

is

o n e -o n e ,a n d for any norm ed space G and any S e L (G ,F ) w ith S(G ) C T (E )

th ere exists exactly one S0 e L (G ,E ) such th a t S = T S 0 (i.e., S can be factorized th ro u g h T from th e left). (6.c) U niversal p ro p e rty of topological suriections and quotients (see Ju n ek [1983]): Let E and F be norm ed spaces and T € L (E ,F ). T hen T is a topological surjection (i.e., open) if and only if T is onto, and for any norm ed space F 0 and any S e L (E ,F 0) w ith K er T c K er S,

there exists exactly one R 0 e L (F ,F 0) such th a t S = R0T (i.e. S can be factorized th ro u g h T from th e rig h t).

W e now describe th e d u ality relations betw een topological injections and topological surjections as follows:

(6.6)

T heorem (P ietsch [1980]). Let E and F be Banach spaces and T

g

L (E ,F ).

T hen

j(T ) = q ( T ') and j ( T ') = q ( T ) .

In p a rtic u la r, T dual o p erato r

T

is a topological injection (resp. m etric injection! if and only if its

is a topological surjection

(resp. m etric surjection); T

surjection (resp . m etric su rje ctio n ) if and only if T

is a topological

is a topological injection (resp. m etric ! injection). C o nsequently, T is an isom orphism (resp. m etric isom orphism ) if and only if T

105

is an isom orphism (resp. m etric isom orphism ).

i P ro o f. T he equality j(T ) = q(T ) clearly follows from

(1)

r||x ||
0 be such th a t

r||x || < ||T x||

(x G E ). F or any

uT U

, , the E

functional f , defined by < T x ,f> = < x ,u '>

( x G E ),

is a lin ear functional on T (E ) (since T is o n e-one). As r||x || < ||T x ||

(x G E ), it follows

th a t

| < T x , f> | = | < x , u ' > | < ||x|| Hu'll < ||x|| < r -‘ ||T x ||;

< T~l- By H ah n —B a n a c h 's extension theorem ,

in o th e r w ords, f is continuous w ith f has an extension y ' € F

w ith ||y '|| = ||f||rr,,px < r 1. In p a rtic u la r,

< x , T y '> = < T x , y '> = < T x , f> = < x , u ' >

th u s u ' = T y ' and n i ' = T ( r y ') G T (U

(x G E)

,), which shows th a t F

rU E C onversely, let r > 0 be such th a t rU E r||x ||

, C t ’(U ,). F

, c T (U ,). F o r any x G E , we have F

= rs u p { | < x , u '> | : u ' G U

, } E


0 be such th a t r U p c T (U p ) .For any v ' E F , we have r ||v '||

= r sup{ | < y , v ' > | : y 6 U p }
| :

X

e Up } < 1
| .

T herefore

l|T g|| = sup{ |< x , T g > | : x e U p } < 1
| < r||y || ||g|| < r||g ||,

107

(6.d)

T h e com position of topological in jectio n s: L et

E, F

an d

G

be Banach

spaces and given th e following two operators: S E ------- ►F

T ►G

T h en T S is a topological injection if and only if (i) (ii)

S :E p ^ S (E ) :

►F is a topological injection, and --------- * G is a topological injection.

D ually, one can verify th e following: (6.e)

T h e com position of topological subjections: L et

E, F

and

G be B anach

spaces an d given th e following two operators: S T E --------►F----------►G

T hen TS is a topological surjection if and only if (i) (ii)

T: F

►G is a topological su rje ctio n ,a n d

^K er T ^

^ ---------* F /K e r T is a topological surjection.

As o th e r im p o rta n t applications of (3.j), we have th e following: (6.f) E xtension pro p erty and lifting p ro p e rty : A B an ach space F is said to have th e extension p ro p e rty (resp. m etric extension p roperty j ,if for an y B anach spaces E E0

, any topological injection (resp. m etric injection),

J : E 0 > --------- ►E

and

and any

S0 6 L (E 0,F) th ere exists an extension S £ L (E .F ) such th a t S0 = SJ (resp. S0 = SJ and ||S|| = ||S0||).

D ually, a B anach space E is said to have the lifting p ro p erty (resp. m etric lifting p roperty)

108

if for any B anach spaces F and F 0, any topological surjection (resp. m etric su rjection and e > 0) Q : F

F 0 and any T 0 G L (E ,F 0) th ere exists a lifting T G L (E ,F ) such th a t T 0 = Q T (resp. T 0 = Q T and ||T || < (1 + ( ) ||T 0||).

(i)

F or any index set

A, the space ^°(A)

has the m etric extension

property; while the space ^ (A ) has th e m etric lifting property. Let E be a B anach space. T he m ap J ^ : E > ---------►(°(XJ ,), defined b E

(ii) by

Jr^X = [ < x,f > ,f G U ,] ^ E is a m etric injection from E in to ^°(U

,).

(x G E ),

(C onsequently, any Banach space is m etrically

E isom orphic m ap

to a subspace of som e B anach space w ith the m etric extension p ro p e rty .) T he

: ^ (U g )

E , defined by

^ E ^ x’ X ^ ^ E ^ =

is a m etric surjection from

^ (U g )

onto

([?x’X ^ ^E^ ^ ^ (U g )),

E .(C onsequently, any Banach space is m etrically

isom orphic to a q uotient space of som e B anach space w ith the m etric lifting p ro p e rty .) (iii)

E -< C°(U

,) if and only if E has the extension property. E

(iv)

E -< ^ (A )

(for som e A ) if and only if E has the lifting property.

Exercises

6—1.

Prove (6.a).

6—2.

Prove (6.b).

6—3.

Prove (6.c).

6^4.

P rove (6.d).

109

6—5.

P ro v e (6.e).

6—6.

P ro v e (6.f).

6—7.

D enote by A a class of B—spaces containing all isom orphic copies of any G 6/A (i.e., if E is a B —space such th at E ~ G (isom orphic) for som e G e / A th e n E 6 A ). Let E a n d F be B -sp a c es and T e L (E ,F ). Show th a t th e follow ing sta te m en ts are eq u iv alen t.

(a)

If M is a closed subspace of E such th a t T J ^ : M — > F is a topological injection, th e n M ( / A . D of E w ith D e/A .

(b)

T has no bounded inverse on subspaces

(c)

T h ere exists no subspace Dof E w ith D e /A such th a t T J ^ : D —» F is a topological injection.

(d)

Let G be a B—space. If there exist topological i n j e c t i o n s 6 L (G ,E ) and R 2 € L (G ,F ) such th a t R 2 = TR^ then G £ lA.

t (a)

D ually, th e following statem en ts are equivalent for any T 6 L (E ,F ): F f F If N is a closed subspace of F such th a t Q ^ T : E — > is open, th e n ! ^ £ I A .

i (c)

T h ere exists no subspace Nq of F w ith

F

/ e /A

/N

such th a t

o

F

F T : E —►

^o

is o

open. i (d)

Let G be a B—space. If there exist open operators

e L (E ,G ) an d S9 e (F ,G ) such

th a t S ^ = S 2T , then G £/A. 6 -8 . (a)

Suppose th a t E, E Q, F and F q are B -spaces. Given tw o o perators SQ € L (E ,F ) and S € L (E ,F ). If ||S0x|| < ||Sx|| (x

€ E)

and F q has th e m etric extension property, th en th ere exists an o p e rato r L 6 L (F ,F ) such th a t SQ = LS (b)

and

||L|| < 1.

D ually given two operators T q e L(E ,F) and T G

L (E ,F ).

If T o (U g ) c T (U ^ ) o

and E q has th e m etric lifting property, then for any e > 0, th e re exists an o p e rato r R e L (E q ,E) such th a t T q = T R (H int : Use (3.2) and Ex. 3—17.)

and

||R || < I + e.

7

Vector Topologies A n a tu ra l generalization of norm ed spaces is vector spaces equipped w ith a topology w hich is induced by a fam ily of sem inorm s instead of one norm . V ector spaces w ith such a topology are special cases of a m ore general class of spaces, called topological vector spaces, as defined by th e following:

(7.1)

D efinition. Let X be a vector space over K. A topology ^ on X is called a

vector topology if it is com patible w ith th e algebraic operations of X in th e following sense: (V T1) the m ap (x,y) ------- ►x + y : X x X -------►X is continuous; (V T 2) th e m ap (A,y) ------- ►Ay : K* X -------►X is continuous;

A vector space equipped w ith a vector topology is called a topological vector space (abbreviated

TV S). H ereafter we shall generally denote a

TVS

by

(X ,^ )

or

X^> or

sim ply by X if the vector topology on X does not require any special notatio n ; and th e term local base,

denoted by ^

, will alw ays m ean a local base at 0 for

Tw o TV S X and Y over K are said to be topologically isom orphic, denoted by X ^ Y, if th ere exists a bijective linear m ap

T :X

►Y which is a hom eom orphism ; T

is referred to as a topological isom orphism from X onto Y.

For a TVS X, th e whole topological stru c tu re of X is determ ined by a local base (hence a linear m ap

T :X

►Y

is continuous if and only if it is continuous a t 0), as

shown by th e following sim ple, bu t im p o rta n t result which is sim ilar to (2.a).

(7.a) H om eom orphism of tra n s la tio n : Let 0 i r 0 £ K , th e tran sla tio n ,d efin ed by

110

X

be a TVS. For an y

x0 e X

and

Ill

y

►x 0 + r 0y

(for all y 6 X),

is a h om eom orphism from X onto X.

V e cto r topologies can be characterized by m eans of a local base as show n by th e follow ing re su lt.

(7.2)

T heorem . In a TVS X^>, th ere exists a local base

^

whose m em bers have

th e follow ing p ro p e rtie s: (N S l) every m em ber in (NS2) for anv V g ^

is circled and a b so rb in g ; th ere is an W g ^

such th a t W 4- W C V.

C o n v ersely , if U is a filter base on X (i.e., any elem ent in for any V l7 V 2 g % there is an V G ^

%C is n o n -e m p ty and

w ith V c Vi fl V 2 ) w hich satisfies (N S l) and (NS2),

th e n th ere e xists a unique vector topology 7 o n

X such th a t %C is a local base at 0 for J.

P ro o f. T he c o n tin u ity of the m ap (A ,x )-------►Ax a t (0,0) ensures th a t th e fam ily of all circled 0—neighbourhoods in X is a local base at 0. [For any 0—neighbourhood W in X th ere is a 0—neighbourhood

U and

8 > 0 such th a t AU c W

(for all

|A | < 8), hence the

set V = U{ AU : | A | < 8 } is a circled 0—neighbourhood w ith V c W .] F o r any fixed x 0e X, th e c o n tin u ity of

(A,x0) -------►Ax0

p a rtic u la r, every V G ^ )

at A = 0

ensures th a t every 0—neighbourhood (in

is absorbing.

B y (V T l) ,th e co n tin u ity of the m ap

( x ,y ) --------- ►x 4- y a t (0,0) ensures th a t

satisfies th e condition (NS2). C onversely, we first define a topology y on X by se ttin g 3r= { < p / G c X : x e G

im plies x 4- V C G (for som e

C learly

J

is th e unique topology on X such

th a t

V G U) } U {

a t 0 consisting of

closed, circled and absorbing sets. (7.3)

D efinition.

A vector topology

& on

X

is said to be locally convex if

&

113

ad m its a local base at

0

consisting of convex sets. A vector space equipped w ith a

H ausdorff locally convex topology is called a locally convex space (a b b re v iate d LCS). L ocally convex topologies can be characterized by m eans of local base as shown by th e follow ing resu lt.

(7.4)

T h eo rem . Let

9 be a locally convex topology on X. T h en th ere exists a local

base U y w hose m em bers have the following p ro p e rties: (LC1) every m em ber in %Cy is absolutely convex and a b so rb in g ; an d A > 0 then AV e U y .

(LC2) if V 6 U

C onversely, if

is a filter base on

X

which satisfies conditions

(L C 2), th e n th e re exists a unique locally convex topology

(LC1)

9 on X such th a t

and

U is a local

base a t 0 for 9 .

P roof.

T here

0—neighbourhoods. If W C V, hence circled;

exists,

by

(7.3),

a

local

base

JC

consisting

V e JC , by (7.2) there is a circled 0—neighbourhood

W

of convex such th a t

W C co W C V, thus co W is a convex 0—n eighbourhood w hich is obviously

consequently,

9

adm its a local base

V

consisting of absolutely

convex

0—neighbourhoods. Now th e family, defined by %Cy = { AW : W G V, A > ° },

has th e required properties. C onversely, th e assertion follows from (7.2) and (7.3).

Locally convex topologies can be determ ined by a fam ily of sem inorm s. T o do this, we req u ire th e following term inology and result (see (7.b) below ). A fam ily IP of sem inorm s on X is s a tu ra te d if m ax Pj e IP whenever p 4 G IP (i = 1,2,• • -n), 1 < i< n

114

w here ( m ax pj) x = m ax Pj(x) 1< i