Functional Analysis And Infinite-Dimensional Geometry

Mari´anFabian2 Petr Habala13 Petr H´ajek12 Vicente Montesinos Santaluc´ıa4 Jan Pelant2 V´aclav Zizler12 1 Department of Mathematics, University of Alberta, Edmonton 2 Mathematical Institute, Czech Academy of Sciences, Prague 3 Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, Prague 4 Departamento de Matem´aticaAplicada, Universidad Polit´ecnicade Valencia Preface

Preface Banach spaces provide a framework for linear and nonlinear functional analysis, operator theory, abstract analysis, probability, optimization and other branches of mathematics. This book is intended as an introduction to linear functional analysis and to some parts of infinite-dimensional Banach space theory. The first seven chapters are directed mainly to undergraduate and graduate students. We have strived to make the text easily readable and as self-contained as possible. In particular, we proved many basic facts that are considered “folk- lore”. An important part of the text is a large number of exercises with detailed hints for their solution. They complement the material in the chapters and contain many important results. The last five chapters introduce the reader to selected topics in the theory of Banach spaces related to smoothness and topology. This part of the book is intended as an introduction to and a complement of existing books on the subject ([BeLi], [DGZ3], [Dis1], [Dis2], [Fab], [JoL3], [LiT2], [Phe2], [Woj]). Some material is presented here for the first time in a monograph form.

The text is based on graduate courses taught at the University of Alberta in Edmonton in the years 1984–1997. These courses were also taken by many senior students in the Honors undergraduate program in Edmonton. As a prerequisite, basic courses in calculus and linear algebra should be enough. For the most part, Royden’s book [Roy] should be sufficient. The chapters are best read consecutively. However: — Chapter 4 as well as the latter part of Chapter 3 (James boundaries) can be omitted in the case of a more elementary functional analysis course. Chapter 4 is used only marginally in Chapters 8–10. — The spectral theory (Chapter 7) can be approached after the first two chapters and the beginning of Chapter 3 were covered; it is not needed in latter chapters. The book can serve as a textbook for the following types of courses in functional analysis: 1. Graduate two-semester course: Chapters 1–9. 2. Graduate one-semester course: Chapters 1–3, 5, and 6 or 7.

5 3. Graduate one-semester advanced course: Chapters 8–10 or 11, 12. 4. Undergraduate first course in functional analysis: Chapters 1–3 and a part of Chapter 7. 5. Undegraduate second course in functional analysis: Chapters 4–6, Chapter 8 and 10. The first three chapters together with Chapter 7 can be used in service courses for students of probability, physics, or engineering.

The principal part of the text was prepared at the Department of Mathemat- ics, University of Alberta in Edmonton. Each author spent some time at this department. Habala and H´ajekobtained their PhD degrees there and Zizler was a faculty member there. We all thank this department for excellent work- ing conditions. We also thank our present home institutions for enabling us to finalize the book. We are indebted to the grant agencies in Canada, the Czech Republic, Germany, Spain and the U.S. for supporting our research in Banach space theory over the years. We are grateful to our colleagues and students for many helpful discussions. Our special thanks go to Jon Borwein, Gilles Godefroy, Jiˇr´ıJel´ınek,Kamil John, L´opez Pellicer, Jos´eOrihuela, Nicole Tomczak-Jaegermann, Jon Vanderwerff, and Dirk Werner. We also thank our colleagues that allowed us to include some of their recent unpublished results. We thank Marion Benedict for her excellent typing of the first version of the manuscript and the staff of Springer-Verlag for their efficient work. Above all, we are deeply indebted to our wives for their support and encour- agement.

We would be glad if this book inspired some young mathematicians to choose Banach spaces as their field of interest, and hope that students and researchers in Banach space theory will find the text useful. We wish the readers a pleasant time spent over this book.

Prague and Valencia Summer 2000 The authors

6 Contents

Preface 5

1 Basic Concepts in Banach Spaces 1 H¨olderand Minkowski inequalities, classical spaces C[0, 1], `p, c0, Lp[0, 1], ...... 2 Operators, quotient spaces, finite-dimensional spaces, Riesz’s lemma, separability ...... 11 Hilbert spaces, orthonormal bases, `2 ...... 17 Exercises ...... 23

2 Hahn-Banach and Banach Open Mapping Theorems 38 Hahn-Banach extension and separation theorems ...... 39 Duals of classical spaces ...... 45 Banach open mapping theorem, closed graph theorem, dual operators 51 Exercises ...... 54

3 Weak Topologies 66 Weak and weak star topology, Banach-Steinhaus uniform boundedness principle, Alaoglu’s and Goldstine’s theorem, reflexivity . . . . . 67 Extreme points, Krein-Milman theorem, James boundary, Ekeland’s variational principle, Bishop-Phelps theorem ...... 79 Exercises ...... 91

4 Locally Convex Spaces 112 Local bases, bounded sets, metrizability and normability, finite-dimensional spaces, distributions ...... 113 Bipolar theorem, Mackey topology ...... 122 Representation and compactness: Carath´eodory and Choquet repre- sentation, Banach-Dieudonn´e,Eberlein-Smulian,ˇ Kaplansky the- orems, Banach-Stone theorem ...... 127 Exercises ...... 136

7 5 Structure of Banach Spaces 143 Projections and complementability, Auerbach bases ...... 143 Separable spaces as subspaces of C[0, 1] and quotients of `1, Sobczyk’s theorem, Schur’s property of `1 ...... 147 Exercises ...... 153

6 Schauder Bases 167 Shrinking and boundedly complete bases, reflexivity, Mazur’s basic sequence theorem, small perturbation lemma ...... 171 Bases in classical spaces: block basis sequences, PeÃlczy´nski’sdecompo- sition method and subspaces of `p, Pitt’s theorem, Khintchine’s inequality and subspaces of Lp ...... 178 Unconditional bases, James’s theorem on containment of `1 and c0, James’s space J, Bessaga-PeÃlczy´nskitheorem ...... 186 Markushevich bases: existence for separable spaces, extension prop- erty, Johnson’s and Plichko’s result on `∞ ...... 194 Exercises ...... 197

7 Compact Operators on Banach Spaces 209 Compact operators and finite rank operators, Fredholm operators Fred- holm alternative ...... 209 Spectral theory: eigenvalues, spectrum, resolvent, eigenspaces . . . . . 216 Self-adjoint operators, spectral theory of compact self-adjoint and com- pact normal operators ...... 223 Fixed points: Banach’s contraction principle, non-expansive mappings, Ryll-Nardzewski theorem, Brouwer’s and Schauder’s theorems, invariant subspaces ...... 234 Exercises ...... 238

8 Differentiability of Norms 249 Smulian’sˇ dual test, Kadec’s Fr´echet-smooth renorming of spaces with separable dual, Fr´echet differentiability of convex functions . . . 251 Extremal structure, Lindenstrauss’ result on strongly exposed points and norm attaining operators ...... 264 Exercises ...... 272

9 Uniform Convexity 294 Uniform convexity and uniform smoothness, `p spaces ...... 294 Finite representability, local reflexivity, superreflexive spaces and En- flo’s renorming, Kadec’s and Gurarii-Gurarii-James theorems . . 300 Exercises ...... 315

8 10 Smoothness and Structure 323 Variational principles (smooth and compact), subdifferential, Stegall’s variational principle ...... 324 Smooth approximation: partitions of unity ...... 338 Lipschitz homeomorphisms, Aharoni’s embeddings into c0, Heinrich- Mankiewicz results on linearization of Lipschitz maps ...... 341 Homeomorphisms: Mazur’s theorem on `p, Kadec’s theorem ...... 346 Smoothness in `p, Hilbert spaces ...... 350 Countable James boundary and saturation by c0 ...... 353 Exercises ...... 357

11 Weakly Compactly Generated Spaces 368 Projectional resolutions, injections into c0(Γ), Eberlein compacts, em- bedding into a reflexive space, locally uniformly rotund and smooth renormings ...... 369 Weakly compact operators, Davis-Figiel-Johnson-PeÃlczy´nskifactoriza- tion, absolutely summing operators, Pietsch factorization, Dunford- Pettis property ...... 382 Quasicomplements ...... 388 Exercises ...... 390

12 Topics in Weak Topology 398 Eberlein compacts, metrizable subspaces ...... 399 Uniform Eberlein compacts, scattered compacts ...... 405 Weakly Lindel¨ofspaces, property C ...... 414 Corson compacts, weak pseudocompactness in Banach spaces, (BX , w) Polish ...... 420 Exercises ...... 428

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462 Index

annihilator yol and Yol, 40, 55, 58, bump. See function 93, 148, 149 (Bx, 1\·11),2,14 (Bx, w), 73, 75, 414, 415 Banach limit, 62 (Bx, w*), 71-73, 319, 365, 395, basis 409-412 algebraic, 34, 191 Auerbach, 139, 164 cardinality card(A), 23, 403 bimonotone, 191 closure ~.~-M , M ,M, 64 block, 172 compact boundedly complete, 166-168, 182, Corson, 409-412, 427, 428 192 countable, 345, 399, 420 constant, 163, 169, 182 Eberlein, 365, 367, 388, 390-393, equivalent, 169-171 409, 417, 419, 420 Hamel (see basis: algebraic) scattered, 398-401, 419, 420 Markushevich, 188-190, 382, uniform Eberlein, 394, 395, 418, 410-412 )~'It 419 \Ja.\d.(vio.l\i2.~ shrinking, 188, 197, 369, 370 complement, 137, 138, 147-149 weakly compact, 364:1>10) algebraic, 137, 147 weakly LindelOf, 411 orthogonal Fol, 17, 18, 138 monotone, 163, 191, 192 quasieomplement, 377 normalized, 163 constant orthonormal, 18, 20, 222 basis, 163, 169 Schauder, 161, 163, 165, 303, 307 unconditional basis, 182 seminormalized, 303 conv(M), 2,22, 85, 92, 104 shrinking, 166-168, 184, 192, 260 convergence summing, 165, 181 in norm -+, 65 uncondition al, 180, 181, 196, 197 pointwise, 66, 68, 86