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BK

New York : Springer, c2011

xiii, 820 s. : il. ; 24 cm

ISBN 978-1-4419-7514-0 (váz.)

CMS books in mathematics, ISSN 1613-5237

V prelimináriích: Canadian Mathematical Society

Obsahuje bibliografii na s. 751-775 a rejstříky

000237200

Contents // 1 Basic Concepts in Banach Spaces ... // 1.1 Basic Definitions... // 1.2 Holder and Minkowski Inequalities, Classical Spaces C[0. I], Č,„co, Lp[0, 1)... // 1.3 Operators, Quotients, Finite-Dimensional Spaces... // 1.4 Hilbert Spaces... // 1.5 Remarks and Open Problems... // Exercises for Chapter 1... // 2 Hahn-Banach and Banach Open Mapping Theorems ... // 2.1 Hahn-Banach Extension and Separation Theorems ... // 2.2 Duals of Classical Spaces... // 2.3 Banach Open Mapping Theorem, Closed Graph Theorem, Dual Operators... // 2.4 Remarks and Open Problems... // Exercises for Chapter 2... // 3 Weak Topologies and Banach Spaces ... // 3.1 Dual Pairs. Weak Topologies... // 3.2 Topological Vector Spaces... // 3.3 Locally Convex Spaces... // 3.4 Polarity... // 3.5 Topologies Compatible with a Dual Pair ... // 3.6 Topologies of Subspaces and Quotients... // 3.7 Weak Compactness... // 3.8 Extreme Points. Krein-Milman Theorem... // 3.9 Representation and Compactness ... // 3.10 The Space of Distributions... // 3.11 Banach Spaces... // 3.11.1 Banach-Stcinhaus Theorem... // 3.11.2 Banach-Dieudonné Theorem... // 1 // I // 3 // 13 // 24 // 29 // 31 // 53 // 54 // 60 // 65 // 68 // 68 // 83 // 83 // 86 // 94 // 98 // 100 // 103 // 104 // 109 // 112 // 115 // 119 // 119 // 122 // x Contenls // 3.11.3 The Bidual Space...125 // 3.11.4 The Completion of a Normed Space...126 // 3.11.5 Separability and Metrizability ...127 // 3.11.6 Weak Compactness...129 // 3.11.7 Reflexivity...129

// 3.11.8 Boundaries...131 // 3.12 Remarks and Open Problems...141 // Exercises for Chapter 3...142 // 4 Schauder Bases ...179 // 4.1 Projections and Complementability, Auerbach Bases...179 // 4.2 Basics on Schauder Bases ...182 // 4.3 Shrinking and Boundedly Complete Bases. Perturbation...187 // 4.4 Block Bases. Bessaga-Pclczyriski Selection Principle...194 // 4.5 Unconditional Bases...200 // 4.6 Bases in Classical Spaces...205 // 4.7 Subspaees of L,, Spaces...213 // 4.8 Markushevich Bases...216 // 4.9 Remarks and Open Problems...218 // Exercises for Chapter 4...220 // 5 Structure of Banach Spaces ...237 // 5.1 Extension of Operators and Lifting...237 // 5.2 Weak Injectivity...250 // 5.2.1 Schur Property...252 // 5.3 Rosenthal’s l\\ Theorem...253 // 5.4 Remarks and Open Problems...264 // Exercises for Chapter 5...267 // 6 Finite-Dimensional Spaces ...291 // 6.1 Finite Representability...291 // 6.2 Spreading Models...294 // 6.3 Complemented Subspaces in Spaces with an Unconditional // Schauder Basis... 298 // 6.4 The Complemented-Subspace Result...309 // 6.5 The John Ellipsoid...312 // 6.6 Kadec-Snobar Theorem...320 // 6.7 Grothendieck’s Inequality ...323 // 6.8 Remarks...325 // Exercises for Chapter 6...326 // 7 Optimization ...331 // 7.1 Introduction...331 // 7.2 Subdifferentials: Šmulyan’s Lemma...336 // Contenis xi // 7.3 Ekeland Principle and Bishop-Phelps Theorem ...351 // 7.4 Smooth Variational Principle...355 // 7.5 Norm-Attaining Operators...359 // 7.6 Michael’s

Selection Theorem...361 // 7.7 Remarks and Open Problems...364 // Exercises for Chapter 7...365 // 8 С’-Smoothness in Separable Spaces ...383 // 8.1 Smoothness and Renormings in Separable Spaces ...383 // 8.2 Equivalence of Separable Asplund Spaces...385 // 8.3 Applications in Convexity ...394 // 8.4 Smooth Approximation ...402 // 8.5 Ranges of Smooth Maps...405 // 8.6 Remarks and Open Problems...408 // Exercises for Chapter 8...410 // 9 Superreflexive Spaces...429 // 9.1 Uniform Convexity and Uniform Smoothness, tp and Lp Spaces. 429 // 9.2 Finite Representability, Superreflexivity...435 // 9.3 Applications...449 // 9.4 Remarks...453 // Exercises for Chapter 9...453 // 10 Higher Order Smoothness...465 // 10.1 Introduction...465 // 10.2 Smoothness in íp ...466 // 10.3 Countable James Boundary...468 // 10.4 Remarks and Open Problems...474 // Exercises for Chapter 10...475 // 11 Dentability and Differentiability ...479 // 11.1 Dentability in X...479 // 11.2 Dentability in X*...486 // 11.3 The Radon-Nikodým Property...490 // 11.4 Extension of Rademacher’s Theorem...504 // 11.5 Remarks and Open Problems...510 // Exercises for Chapter 11...511 // 12 Basics in Nonlinear Geometric Analysis...521 // 12.1 Contractions and Nonexpansive Mappings...521 // 12.2 Brouwer and Schauder Theorems...526 // xii Contents // 12.3 The Homeomorphisms of Convex Compact Sets: Keller’s // Theorem... 533 // 12.3.1 Introduction ...533 // 12.3.2 Elliptically Convex Sets ...535 // 12.3.3 The Spacer...537

// 12.3.4 Compact Elliptically Convex Subsets off 2...538 // 12.3.5 Keller Theorem ...541 // 12.3.6 Applications to Fixed Points...541 // 12.4 Homeomorphisms: Kadec’s Theorem...542 // 12.5 Lipschitz Homeomorphisms ...545 // 12.6 Remarks and Open Problems...559 // Exercises for Chapter 12...561 // 13 Weakly Compactly Generated Spaces ...575 // 13.1 Introduction...575 // 13.2 Projectional Resolutions of the Identity...577 // 13.3 Consequences of the Existence of a Projectional Resolution 581 // 13.4 Renormings of Weakly Compactly Generated Banach Spaces 586 // 13.5 Weakly Compact Operators...591 // 13.6 Absolutely Summing Operators...592 // 13.7 The Dunlbrd-Pettis Property...596 // 13.8 Applications...598 // 13.9 Remarks and Open Problems...602 // Exercises for Chapter 13...603 // 14 Topics in Weak Topologies on Banach Spaces...617 // 14.1 Eberlein Compact Spaces...617 // 14.2 Uniform Eberlein Compact Spaces...622 // 14.3 Scattered Compact Spaces...625 // 14.4 Weakly Lindelöf Spaces. Property С...629 // 14.5 Weak* Topology of the Dual Unit Ball...634 // 14.6 Remarks and Open Problems...642 // Exercises for Chapter 14...643 // 15 Compact Operators on Banach Spaces...657 // 15.1 Compact Operators...657 // 15.2 Spectral Theory...661 // 15.3 Self-Adjoint Operators...668 // 15.4 Remarks and Open Problems...678 // Exercises for Chapter 15...678 // 16 Tensor Products...687 // 16.1 Tensor Products and Their Topologies ...687 // 16.2 Duality of Injective Tensor Products...696 // Contents

// хш // 16.3 Approximation Property and Duality of Spaces of Operators ... 700 // 16.4 The Trace...708 // 16.5 Banach Spaces Without the Approximation Property...711 // 16.6 The Bounded Approximation Property...717 // 16.7 Schauder Bases in Tensor Products...721 // 16.8 Remarks and Open Problems...726 // Exercises for Chapter 16...727 // 17 Appendix...733 // 17.1 Basics in Topology...733 // 17.2 Nets and Fillers...735 // 17.3 Nets and Fillers in Topological Spaces...736 // 17.4 Ultraproducts...737 // 17.5 The Order Topology on the Ordinals...737 // 17.6 Continuity of Set-Valued Mappings ...738 // 17.7 The Cantor Space ...739 // 17.8 Baire’s Great Theorem...741 // 17.9 Polish Spaces...741 // 17.10 Uniform Spaces...741 // 17.1 I Nets and Filters in Uniform Spaces...742 // 17.12 Partitions of Unity...743 // 17.13 Measure and Integral...744 // 17.13.1 Measure...744 // 17.13.2 Integral ...745 // 17.14 Continued Fractions and the Representation of the Irrational // Numbers... 746 // References...751 // Symbol Index...777 // Subject Index...781 // Author Index ...807