Theoretical and Mathematical Physics

The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs inPhysics (TMP) publishes high-level monographs in theoretical and mathematical physics.The change of title to Theoretical and Mathematical Physics (TMP) signals that the series isa suitable publication platform for both the mathematical and the theoretical physicist. Thewider scope of the series is reflected by the composition of the editorial board, comprisingboth physicists and mathematicians.

The books, written in a didactic style and containing a certain amount of elementary back-ground materials, bridge the gap between advanced textbooks and research monographs.They can thus serve as basis for advanced studies, not only for lectures and seminars atgraduate level, but also for scientists entering a field of research.

Editorial Board

W. Beiglböck, Institute of Applied Mathematics, University of Heidelberg, GermanyJ.-P. Eckmann, Department of Theoretical Physics, University of Geneva, SwitzerlandH. Grosse, Institute of Theoretical Physics, University of Vienna, AustriaM. Loss, School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USAS. Smirnov, Mathematics Section, University of Geneva, SwitzerlandL. Takhtajan, Department of Mathematics, Stony Brook University, NY, USAJ. Yngvason, Institute of Theoretical Physics, University of Vienna, Austria

Jiří Blank • Pavel Exner • Miloslav Havlíček

Hilbert Space Operatorsin Quantum Physics

Second Edition

ABC

Jiří Blank†

PragueCzechia

Pavel ExnerDoppler InstituteBřehová 711519 Pragueand Nuclear Physics InstituteCzech Academy of Sciences25068 Řež near PragueCzech Republicexner@ujf.cas.cz

Miloslav HavlíčekDoppler Instituteand Faculty of Nuclear Sciencesand Physical EngineeringCzech Technical UniversityTrojanova 1312000 PragueCzech Republichavlicek@fjfi.cvut.cz

ISBN 978-1-4020-8869-8 e-ISBN 978-1-4020-8870-4

Library of Congress Control Number: 2008933703

All Rights Reservedc© 2008 Springer Science + Business Media B.V.c© 1993, first edition, AIP, Melville, NY

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Charles University

To our wives and daughters

Preface to the second edition

Almost fifteen years later, and there is little change in our motivation. Mathemat-ical physics of quantum systems remains a lively subject of intrinsic interest withnumerous applications, both actual and potential.

In the preface to the first edition we have described the origin of this book rootedat the beginning in a course of lectures. With this fact in mind, we were naturallypleased to learn that the volume was used as a course text in many points of theworld and we gladly accepted the offer of Springer Verlag which inherited the rightsfrom our original publisher, to consider preparation of a second edition.

It was our ambition to bring the reader close to the places where real life dwells,and therefore this edition had to be more than a corrected printing. The field isdeveloping rapidly and since the first edition various new subjects have appeared;as a couple of examples let us mention quantum computing or the major progress inthe investigation of random Schrödinger operators. There are, however, good sourcesin the literature where the reader can learn about these and other new developments.

We decided instead to amend the book with results about new topics whichare less well covered, and the same time, closer to the research interests of one ofus. The main change here are two new chapters devoted to quantum waveguidesand quantum graphs. Following the spirit of this book we have not aspired to fullcoverage — each of these subjects would deserve a separate monograph — but wehave given a detailed enough exposition to allow the interested reader to follow (andenjoy) fresh research results in this area. In connection with this we have updatedthe list of references, not only in the added chapters but also in other parts of thetext in the second part of the book where we found it appropriate.

Naturally we have corrected misprints and minor inconsistencies spotted in thefirst edition. We thank the colleagues who brought them to our attention, in particu-lar to Jana Stará, who indicated numerous improvements. As with the first edition,we have asked a native speaker to try to remove the foreign “accent” from ourwriting; we are grateful to Mark Harmer for accepting this role.

Prague, December 2007 Pavel ExnerMiloslav Havĺıček

vii

Preface

Relations between mathematics and physics have a long and entangled tradition.In spite of repeated clashes resulting from the different aims and methods of thetwo disciplines, both sides have always benefitted. The place where contacts aremost intensive is usually called mathematical physics, or if you prefer, physicalmathematics. These terms express the fact that mathematical methods are neededhere more to understand the essence of problems than as a computational tool, andconversely, the investigated properties of physical systems are also inspiring fromthe mathematical point of view.

In fact, this field does not need any advocacy. When A. Wightman remarkeda few years ago that it had become “socially acceptable”, it was a pleasant under-statement; by all accounts, mathematical physics is flourishing. It has long left theadolescent stage when it cherished only oscillating strings and membranes; nowadaysit has built synapses to almost every part of physics. Evidence that the discipline isdeveloping actively is provided by the fruitful oscillation between the investigationof particular systems and synthetizing generalizations, as well as by discoveries ofnew connections between different branches.

The drawback of this rapid development is that it has become virtually impos-sible to write a textbook on mathematical physics as a single topic. There are, ofcourse, books which cover a wide range of problems, some of them indeed monu-mental, but even they are like cities which govern the territory while watching thefrontier slowly moving towards the gray distance. This is simply the price we have topay for the flood of ideas, concepts, tools, and results that our science is producing.

It was not our aim to write a poor man’s version of some of the big textbooks.What we want is to give students basic information about the field, by which wemean an amount of knowledge that could constitute the basis of an intensive one–year course for those who already have the necessary training in algebra and analysis,as well as in classical and quantum mechanics. If our exposition should kindle interestin the subject, the student will be able, after taking such a course, to read specializedmonographs and research papers, and to discover a research topic to his or hertaste. We have mentioned that the span of the contemporary mathematical physicsis vast; nevertheless the cornerstone remains where it was laid by J. von Neumann,H. Weyl, and the other founding fathers, namely in regions connected with quantumtheory. Apart from its importance for fundamental problems such as the constitutionof matter, this claim is supported by the fact that quantum theory is gradually

ix

x Preface

becoming a basis for most branches of applied physics, and has in this way enteredour everyday life.

The mathematical backbone of quantum physics is provided by the theory oflinear operators on Hilbert spaces, which we discuss in the first half of this book.Here we follow a well–trodden path; this is why references in this part aim mostly atstandard book sources, even for the few problems which maybe go beyond the stan-dard curriculum. To make the exposition self–contained without burdening the maintext, we have collected the necessary information about measure theory, integration,and some algebraic notions in the appendices.

The physical chapters in the second half are not intended to provide a self–contained exposition of quantum theory. As we have remarked, we suppose that thereader has background knowledge up to the level of a standard quantum mechan-ics course; the present text should rather provide new insights and help to reach adeeper understanding. However, we attempt to describe the mathematical founda-tions of quantum theory in a sufficiently complete way, so that a student comingfrom mathematics can start his or her way into this part of physics through our book.

In connection with the intended purpose of the text, the character of referencingchanges in the second part. Though the material discussed here is with a few excep-tions again standard, we try in the notes to each chapter to explain extensions ofthe discussed results and their relations to other problems; occasionally we have settraps for the reader’s curiosity. The notes are accompanied by a selective but quitebroad list of references, which map ways to the areas where real life dwells.

Each chapter is accompanied by a list of problems. Solving at least some ofthem in full detail is the safest way for the reader to check that he or she has indeedmastered the topic. The problem level ranges from elementary exercises to fairlycomplicated proofs and computations. We have refrained from marking the moredifficult ones with asterisks because such a classification is always subjective, andafter all, in real life you also often do not know in advance whether it will take youan hour or half a year to deal with a given problem.

Let us add a few words about the history of the book. It originates from coursesof lectures we have given in different forms during the past two decades at CharlesUniversity and the Czech Technical University in Prague. In the 1970s we preparedseveral volumes of lecture notes; ten years later we returned to them and rewrotethe material into a textbook, again in Czech. It was prepared for publication in1989, but the economic turmoil which inevitably accompanied the welcome changesdelayed its publication, so that it appeared only recently.

In the meantime we suffered a heavy blow. Our friend and coauthor, Jǐŕı Blank,died in February 1990 at the age of 50. His departure reminded us of the bittertruth that we usually are able to appreciate the real value of our relationships withfellow humans only after we have lost them. He was always a stabilizing elementof our triumvirate of authors, and his spirit as a devoted and precise teacher is feltthroughout this book; we hope that in this indirect way his classes will continue.

Preparing the English edition was therefore left to the remaining two authors.It has been modified in many places. First of all, we have included two chapters and

Preface xi

some other material which was prepared for the Czech version but then left out dueto editorial restrictions. Though the aim of the book is not to report on the presentstate of research, as we have already remarked, the original manuscript was finishedfour years ago and we felt it was necessary to update the text and references insome places. On the other hand, since the audience addressed by the English text isdifferent — and is equipped with different libraries — we decided to rewrite certainparts from the first half of the book in a more condensed form.

One consequence of these alterations was that we chose to do the translationourselves. This decision contained an obvious danger. If you write in a languagewhich you did not master during your childhood, the result will necessarily containsome unwanted comical twists reminiscent of the famous character of Leo Rosten.We are indebted to P. Moylan and, in particular, to R. Healey, who have read thetext and counteracted our numerous petty attacks against the English language;those clumsy expressions that remain are, of course, our own.

There are many more people who deserve our thanks: coauthors of our researchpapers, colleagues with whom we have had the pleasure of exchanging ideas, andsimply friends who have supported us during difficult times. We should not forgetabout students in our courses who have helped just by asking questions; some ofthem have now become our colleagues. In view of the book complex history, thelist should be very long. We prefer to thank all of them anonymously. However,since every rule should have an exception, let us name J. Dittrich, who read themanuscript and corrected numerous mistakes. Last but not least we want to thankour wives, whose patience and understanding made the writing of this book possible.

Prague, July 1993 Pavel ExnerMiloslav Havĺıček

Contents

Preface to the second edition vii

Preface ix

1 Some notions from functional analysis 1

1.1 Vector and normed spaces . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Metric and topological spaces . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Banach spaces and operators on them . . . . . . . . . . . . . . . . . . 15

1.6 The principle of uniform boundedness . . . . . . . . . . . . . . . . . . 23

1.7 Spectra of closed linear operators . . . . . . . . . . . . . . . . . . . . 25

Notes to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Hilbert spaces 41

2.1 The geometry of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 41

2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3 Direct sums of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . 50

2.4 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Notes to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Bounded operators 63

3.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Hermitean operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Unitary and isometric operators . . . . . . . . . . . . . . . . . . . . . 72

3.4 Spectra of bounded normal operators . . . . . . . . . . . . . . . . . . 74

3.5 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.6 Hilbert–Schmidt and trace–class operators . . . . . . . . . . . . . . . 81

Notes to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xiii

xiv Contents

4 Unbounded operators 934.1 The adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 Closed operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3 Normal operators. Self–adjointness . . . . . . . . . . . . . . . . . . . 1004.4 Reducibility. Unitary equivalence . . . . . . . . . . . . . . . . . . . . 1054.5 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.6 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.7 Self–adjoint extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.8 Ordinary differential operators . . . . . . . . . . . . . . . . . . . . . . 1264.9 Self–adjoint extensions of differential operators . . . . . . . . . . . . . 133Notes to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5 Spectral theory 1515.1 Projection–valued measures . . . . . . . . . . . . . . . . . . . . . . . 1515.2 Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.3 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.4 Spectra of self–adjoint operators . . . . . . . . . . . . . . . . . . . . . 1715.5 Functions of self–adjoint operators . . . . . . . . . . . . . . . . . . . . 1765.6 Analytic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.7 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.8 Spectral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 1875.9 Groups of unitary operators . . . . . . . . . . . . . . . . . . . . . . . 191Notes to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6 Operator sets and algebras 2056.1 C∗–algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.2 GNS construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2086.3 W ∗–algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2136.4 Normal states on W ∗–algebras . . . . . . . . . . . . . . . . . . . . . . 2216.5 Commutative symmetric operator sets . . . . . . . . . . . . . . . . . 2276.6 Complete sets of commuting operators . . . . . . . . . . . . . . . . . 2326.7 Irreducibility. Functions of non-commuting operators . . . . . . . . . 2356.8 Algebras of unbounded operators . . . . . . . . . . . . . . . . . . . . 239Notes to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7 States and observables 2517.1 Basic postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2517.2 Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.3 Mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2647.4 Superselection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2687.5 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Contents xv

7.6 The algebraic approach . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Notes to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

8 Position and momentum 293

8.1 Uncertainty relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

8.2 The canonical commutation relations . . . . . . . . . . . . . . . . . . 299

8.3 The classical limit and quantization . . . . . . . . . . . . . . . . . . . 306

Notes to Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

9 Time evolution 317

9.1 The fundamental postulate . . . . . . . . . . . . . . . . . . . . . . . . 317

9.2 Pictures of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

9.3 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

9.4 The Feynman integral . . . . . . . . . . . . . . . . . . . . . . . . . . 330

9.5 Nonconservative systems . . . . . . . . . . . . . . . . . . . . . . . . . 334

9.6 Unstable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

Notes to Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

10 Symmetries of quantum systems 357

10.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

10.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

10.3 General space–time transformations . . . . . . . . . . . . . . . . . . . 370

Notes to Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

11 Composite systems 379

11.1 States and observables . . . . . . . . . . . . . . . . . . . . . . . . . . 379

11.2 Reduced states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

11.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

11.4 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

11.5 Separation of variables. Symmetries . . . . . . . . . . . . . . . . . . . 392

Notes to Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

12 The second quantization 403

12.1 Fock spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

12.2 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . 408

12.3 Systems of noninteracting particles . . . . . . . . . . . . . . . . . . . 413

Notes to Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

xvi Contents

13 Axiomatization of quantum theory 425

13.1 Lattices of propositions . . . . . . . . . . . . . . . . . . . . . . . . . . 425

13.2 States on proposition systems . . . . . . . . . . . . . . . . . . . . . . 430

13.3 Axioms for quantum field theory . . . . . . . . . . . . . . . . . . . . 434

Notes to Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

14 Schrödinger operators 443

14.1 Self–adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

14.2 The minimax principle. Analytic perturbations . . . . . . . . . . . . . 448

14.3 The discrete spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 454

14.4 The essential spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 462

14.5 Constrained motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

14.6 Point and contact interactions . . . . . . . . . . . . . . . . . . . . . . 474

Notes to Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

15 Scattering theory 491

15.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

15.2 Existence of wave operators . . . . . . . . . . . . . . . . . . . . . . . 498

15.3 Potential scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

15.4 A model of two–channel scattering . . . . . . . . . . . . . . . . . . . 510

Notes to Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

16 Quantum waveguides 527

16.1 Geometric effects in Dirichlet stripes . . . . . . . . . . . . . . . . . . 527

16.2 Point perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

16.3 Curved quantum layers . . . . . . . . . . . . . . . . . . . . . . . . . . 539

16.4 Weak coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

Notes to Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

17 Quantum graphs 561

17.1 Admissible Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . 561

17.2 Meaning of the vertex coupling . . . . . . . . . . . . . . . . . . . . . 566

17.3 Spectral and scattering properties . . . . . . . . . . . . . . . . . . . . 570

17.4 Generalized graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

17.5 Leaky graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

Notes to Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

Contents xvii

A Measure and integration 595A.1 Sets, mappings, relations . . . . . . . . . . . . . . . . . . . . . . . . . 595A.2 Measures and measurable functions . . . . . . . . . . . . . . . . . . . 598A.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601A.4 Complex measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605A.5 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

B Some algebraic notions 609B.1 Involutive algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609B.2 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612B.3 Lie algebras and Lie groups . . . . . . . . . . . . . . . . . . . . . . . 614

References 617

List of symbols 647

Index 651

Chapter 1

Some notions from functionalanalysis

1.1 Vector and normed spaces

The notion of a vector space is obtained by axiomatization of the properties ofthe three–dimensional space of Euclidean geometry, or of configuration spaces ofclassical mechanics. A vector (or linear) space V is a set {x, y, . . . } equippedwith the operations of summation, [x, y] �→ x + y ∈ V , and multiplication by acomplex or real number α, [α, x] �→ αx ∈ V , such that

(i) The summation is commutative, x + y = y + x, and associative, (x + y) + z =x+(y +z). There exist a zero element 0 ∈ V , and an inverse element −x ∈ V,to any x ∈ V so that x + 0 = x and x + (−x) = 0 holds for all x ∈ V .

(ii) α(βx) = (αβ)x and 1x = x.

(iii) The summation and multiplication are distributive, α(x + y) = αx + αy and(α + β)x = αx + βx.

The elements of V are called vectors. The set of numbers (or scalars) in the definitioncan be replaced by any algebraic field F . Then we speak about a vector space over F ,and in particular, about a complex and real vector space for F = C, R, respectively.A vector space without further specification in this book always means a complexvector space.

1.1.1 Examples: (a) The space Cn consists of n–tuples of complex numbers withthe summation and scalar multiplication defined componentwise. In the sameway, we define the real space Rn.

(b) The space �p, 1 ≤ p ≤ ∞, of all complex sequences X := {ξj}∞j=1 such that∑∞j=1 |ξj|p

2 1 Some notions from functional analysis

(c) The space C(J) of continuous complex functions on a closed interval J ⊂ Rwith (αf +g)(x) := αf(x)+g(x). In a similar way, we define the space C(X)of continuous functions on a compact X and spaces of bounded continuousfunctions on more general topological spaces (see the next two sections).

A subspace L ⊂ V is a subset, which is itself a vector space with the sameoperations. A minimal subspace containing a given subset M ⊂ V is called thelinear hull (envelope) of M and denoted as Mlin or lin(M). Vectors x1, . . . , xn ∈ Vare linearly independent if α1x1 + · · · + αnxn = 0 implies that all the numbersα1, . . . , αn are zero; otherwise they are linearly dependent, which means some of themcan be expressed as a linear combination of the others. A set M ⊂ V is linearlyindependent if each of its finite subsets consists of linearly independent vectors.

This allows us to introduce the dimension of a vector space V as a maxi-mum number of linearly independent vectors in V . Among the spaces mentioned inExample 1.1.1, Cn and Rn are n–dimensional (Cn is 2n–dimensional as a real vec-tor space) while the others are infinite–dimensional. A basis of a finite–dimensionalV is any linearly independent set B ⊂ V such that Blin = V ; it is clear thatdim V = n iff V has a basis of n elements. Vector spaces V, V ′ are said tobe (algebraically) isomorphic if there is a bijection f : V → V ′, which is linear,f(αx + y) = αf(x) + f(y). Isomorphic spaces have the same dimension; for finite–dimensional spaces the converse is also true (Problem 3).

There are various ways to construct new vector spaces from given ones. Let usmention two of them:

(i) If V1, . . . , VN are vector spaces over the same field; then we can equip theCartesian product V := V1× · · · × VN with a summation and scalar multipli-cation defined by α[x1, . . . , xN ]+ [y1, . . . , yN ] := [αx1 +y1, . . . , αxN +yN ]. Theaxioms are obviously satisfied; the resulting vector space is called the directsum of V1, . . . , VN and denoted as V1 ⊕ · · · ⊕ VN or

∑⊕j Vj. The same term

and symbols are used if V1, . . . , VN are subspaces of a given space V suchthat each x ∈ V has a unique decomposition x = x1 + · · ·+ xN , xj ∈ Vj.

(ii) If W is a subspace of a vector space V , we can introduce an equivalencerelation on V by x ∼ y if x−y ∈W . Defining the vector–space operations onthe set Ṽ of equivalence classes by αx̃+ ỹ := (αx+y)̃ for some x ∈ x̃, y ∈ ỹ,we get a vector space, which is called the factor space of V with respect toW and denoted as V/W .

1.1.2 Example: The space Lp(M, dµ) , p ≥ 1, where µ is a non–negative measure,consists of all measurable functions f : M → C satisfying

∫M|f |pdµ < ∞ with

pointwise summation and scalar multiplication — cf. Appendix A.3. The subsetL0 ⊂ Lp of the functions such that f(x) = 0 for µ–almost all x ∈ M is easilyseen to be a subspace; the corresponding factor space Lp(M, dµ) := Lp(M, dµ)/L0is then formed by the classes of µ–equivalent functions.

1.1 Vector and normed spaces 3

A map f : V → C on a vector space V is called a functional; if it maps into thereals we speak about a real functional. A functional f is additive if f(x + y) = f(x)+f(y) holds for all x, y ∈ V , and homogeneous if f(αx) = αf(x) or antihomogeneousif f(αx) = ᾱf(x) for x ∈ V, α ∈ C. An additive (anti)homogeneous functional iscalled (anti)linear. A real functional p is called a seminorm if p(x + y) ≤ p(x)+p(y)and p(αx) = |α|p(x) holds for any x, y ∈ V, α ∈ C; this definition implies that pmaps V into R+ and |p(x)−p(y)| ≤ p(x−y). The following important result isvalid (see the notes to this chapter).

1.1.3 Theorem (Hahn–Banach): Let p be a seminorm on a vector space V . Anylinear functional f0 defined on a subspace V0 ⊂ V and fulfilling |f0(y)| ≤ p(y) forall y ∈ V0 can be extended to a linear functional f on V such that |f(x)| ≤ p(x)holds for any x ∈ V .

A map F := V × · · · × V → C is called a form, in particular, a real form if itsrange is contained in R. A form F : V × V → C is bilinear if it is linear in botharguments, and sesquilinear if it is linear in one of them and antilinear in the other.Most frequently we shall drop the adjective when speaking about sesquilinear forms;we shall use the “physical” convention assuming that such a form is antilinear inthe left argument. For a given F we define the quadratic form (generated by F ) byqF : qF (x) = F (x, x); the correspondence is one–to–one as the polarization formula

F (x, y) =1

4

(qF (x+y)− qF (x−y)

)− i

4

(qF (x+iy)− qF (x−iy)

)

shows. A form F is symmetric if F (x, y) = F (y, x) for all x, y ∈ V ; it is positiveif qF (x) ≥ 0 for any x ∈ V and strictly positive if, in addition, F (x) = 0 holdsfor x = 0 only. A positive form is symmetric (Problem 6) and fulfils the Schwarzinequality,

|F (x, y)|2 ≤ qF (x)qF (y) .A norm on a vector space V is a seminorm ‖ · ‖ such that ‖x‖ = 0 holds

iff x = 0. A pair (V, ‖ · ‖) is called a normed space; if there is no danger ofmisunderstanding we shall speak simply about a normed space V .

1.1.4 Examples: (a) In the spaces Cn and Rn, we introduce

‖x‖∞ := max1≤j≤n

|ξj| and ‖x‖p :=(

n∑

j=1

|ξj|p)1/p

, p ≥ 1 ,

for x = {ξ1, . . . , ξn}; the norm ‖ · ‖2 on Rn is often also denoted as | · |.Analogous norms are used in �p (see also Problem 8).

(b) In Lp(M, dµ), we introduce

‖f‖p :=(∫

M

|f |pdµ)1/p

.

4 1 Some notions from functional analysis

The relation ‖f‖p =0 implies f(x)=0 µ–a.e. in M , so f is the zero elementof Lp(M, dµ). If we speak about Lp(M, dµ) as a normed space, we alwayshave in mind this natural norm though it is not, of course, the only possibility.If the measure µ is discrete with countable support, Lp(M, dµ) is isomorphicto �p and we recover the norm ‖ · ‖p of the previous example.

(c) By L∞(M, dµ) we denote the set of classes of µ–equivalent functions f :M → C, which are bounded a.e., i.e., there is c > 0 such that |f(x)| ≤ cfor µ–almost all x ∈ M . The infimum of all such numbers is denoted assup ess |f(x)|. We can easily check that L∞(M, dµ) is a vector space andf �→ ‖f‖∞ := sup ess x∈M |f(x)| is a norm on it.

(d) The space C(X) can be equipped with the norm ‖f‖∞ := supx∈X |f(x)|.

A strictly positive sesquilinear form on a vector space V is called an inner (orscalar) product. In other words, it is a map (·, ·) from V × V to C such thatthe following conditions hold for any x, y, z ∈ V and α ∈ C:

(i) (x, αy+z) = α(x, y) + (x, z)

(ii) (x, y) = (y, x)

(iii) (x, x) ≥ 0 and (x, x)=0 iff x=0

A vector space with an inner product is called a pre–Hilbert space. Any such spaceis at the same time a normed space with the norm ‖x‖ :=

√(x, x); the Schwarz

inequality then assumes the form

|(x, y)| ≤ ‖x‖ ‖y‖ .

The above norm is said to be induced by the inner product. Due to conditions (i)and (ii) it fulfils the parallelogram identity,

‖x+y‖2 + ‖x−y‖2 = 2‖x‖2 + 2‖y‖2 ;

on the other hand, it allows us to express the inner product by polarization,

(x, y) =1

4

(‖x+y‖2 − ‖x−y‖2

)− i

4

(‖x+iy‖2 − ‖x−iy‖2

).

These properties are typical for a norm induced by an inner product (Problem 11).Vectors x, y of a pre–Hilbert space V are called orthogonal if (x, y) = 0. A

vector x is orthogonal to a set M if (x, y) = 0 holds for all y ∈ M ; the set of allsuch vectors is denoted as M⊥ and called the orthogonal complement to M . Inner–product linearity implies that it is a subspace, (M⊥)lin = M

⊥, with the followingsimple properties

(Mlin)⊥ = M⊥ , Mlin ⊂ (M⊥)⊥ , M ⊂ N ⇒ M⊥ ⊃ N⊥ .

1.2 Metric and topological spaces 5

A set M of nonzero vectors whose every two elements are orthogonal is called anorthogonal set; in particular, M is orthonormal if ‖x‖ = 1 for each x ∈ M . Anyorthonormal set is obviously linearly independent, and in the opposite direction wehave the following assertion, the proof of which is left to the reader.

1.1.5 Theorem (Gram-Schmidt): Let N be an at most countable linearly inde-pendent set in a pre–Hilbert space V , then there is an orthonormal set M ⊂ V ofthe same cardinality such that Mlin = Nlin.

1.2 Metric and topological spaces

A metric on a set X is a map � : X ×X → [0,∞), which is symmetric, �(x, y) =�(y, x), �(x, y) = 0 iff x = y, and fulfils the triangle inequality,

�(x, z) ≤ �(x, y) + �(y, z) ,

for any x, y, z ∈ X; the pair (X, �) is called a metric space (we shall again forsimplicity often use the symbol X only). If X is a normed space, one can definea metric on X by �(x, y) := ‖x−y‖; we say it is induced by the norm (see alsoProblems 15 and 16).

Let us first recall some basic notions and properties of metric spaces. An ε–neighborhood of a point x ∈ X is the open ball Uε(x) := { y ∈ X : �(y, x) < ε}.A point x is an interior point of a set M if there is a Uε(x) ⊂M . A set is open ifall its points are interior points, in particular, any neighborhood of a given point isopen. A union of an arbitrary family of open sets is again an open set; the same istrue for finite intersections of open sets.

The closure M of a set M is the family of all points x ∈ X such that theintersection Uε(x)∩M �= ∅ for any ε > 0. A point x ∈M is called isolated if thereis Uε(x) such that Uε(x)∩M = {x}, otherwise x is a limit (or accumulation) pointof M . The closure points of M which are not interior form the boundary bd M ofM . A set is closed if it coincides with its closure, and M is the smallest closed setcontaining M (cf. Problem 17). In particular, the whole X and the empty set ∅are closed and open at the same time.

A set M is said to be dense in a set N ⊂ X if M ⊃ N ; it is everywheredense if M = X and nowhere dense if X \M is everywhere dense. A metric spacewhich contains a countable everywhere dense set is called separable. An example isthe space Cn with any of the norms of Example 1.1.4a where a dense set is formed,e.g., by n–tuples of complex numbers with rational real and imaginary parts; otherexamples will be given in the next chapter (see also Problem 18).

A sequence {xn} ⊂ X converges to a point x ∈ X if to any Uε(x) there isn0 such that xn ∈ Uε(x) holds for all n > n0. Since any two mutually differentpoints x, y ∈ X have disjoint neighborhoods, each sequence has at most one limit.Sequences can also be used to characterize closure of a set (Problem 17).

Next we recall a few notions related to maps f : X → X ′ of metric spaces. Themap f is continuous at a point x ∈ X if to any U ′ε(f(x)) there is a Uδ(x) such

6 1 Some notions from functional analysis

that f(Uδ(x)) ⊂ U ′ε(f(x)); alternatively we can characterize the local continuityusing sequences (Problem 19). On the other hand, f is (globally) continuous if thepull–back f (−1)(G′) of any open set G′ ⊂ X ′ is open in X.

An important class of continuous maps is represented by homeomorphisms, i.e.,bijections f : X → X ′ such that both f and f−1 are continuous. It is clear thatin this way any family of metric spaces can be divided into equivalence classes. Ahomeomorphism maps, in particular, the family τ of open sets in X bijectivelyonto the family τ ′ of open sets in X ′; we say that homeomorphic metric spacesare topologically equivalent. Such spaces can still differ in metric properties. As anexample, consider the spaces R and (−π

2, π

2) with the same metric �(x, y) := |x−y|;

they are homeomorphic by x �→ arctan x but only the first of them contains un-bounded sets. A bijection f := X → X ′ which preserves the metric properties,�′(f(x), f(y)) = �(x, y), is called isometry; this last named property implies conti-nuity, so any isometry is a homeomorphism.

A homeomorphism f : V → V ′ of normed spaces is called linear homeomor-phism if it is simultaneously an isomorphism. Linearly homeomorphic spaces there-fore also have the same algebraic structure; this is particularly simplifying in thecase of finite dimension (Problem 21). In addition, if the identity ‖f(x)‖V ′ = ‖x‖Vholds for any x ∈ V we speak about a linear isometry.

A sequence {xn} in a metric space X is called Cauchy if to any ε > 0 thereis nε such that �(xn, xm) < ε for all n,m > nε. In particular, any convergentsequence is Cauchy; a metric space in which the converse is also true is calledcomplete. Completeness is one of the basic “nontopological” properties of metricspaces: recall the spaces R and (−π

2, π

2) mentioned above; they are homeomorphic

but only the first of them is complete.

1.2.1 Example: Let us check the completeness of Lp(M, dµ) , p ≥ 1, with a σ–finitemeasure µ. Suppose first µ(M) < ∞ and consider a Cauchy sequence {fn} ⊂ Lp.By the Hölder inequality, it is Cauchy also with respect to ‖ · ‖1, so for any ε > 0there is N(ε) such that ‖fn−fm‖1 < ε for n,m > N(ε). We pick a subsequence,gn := fkn , by choosing k1 := N(2

−1) and kn+1 := max{kn +1, N(2−n−1))}, so‖gn+1−gn‖1 < 2−n, and the functions ϕn := |g1|+

∑n−1�=1 |g�+1−g�| obey

∫

M

ϕndµ ≤ ‖g1‖1 +n−1∑

�=1

2−� < 1 + ‖g1‖1 .

Since they are measurable and form a nondecreasing sequence, the monotone–convergence theorem implies existence of a finite limn→∞ ϕn(x) for µ–a.a. x ∈ M .Furthermore, |gn+p−gn| ≤ ϕn+p−ϕn, so there is a function f which is finite µ-a.e.in M and fulfils f(x) = limn→∞ gn(x). The sequence {gn} has been picked from aCauchy sequence and it is therefore Cauchy also, ‖gn−gm‖p < ε for all n,m > Ñ(ε)for a suitable Ñ(ε). On the other hand, limm→∞ |gn(x)−gm(x)|p = |gn(x)−f(x)|pfor µ–a.a. x ∈ M , so Fatou’s lemma implies ‖gn−f‖p ≤ ε for all n > Ñ(ε); hencef ∈ Lp and limn→∞ ‖fn−f‖p = 0 (Problem 24).

1.2 Metric and topological spaces 7

If µ is σ–finite and µ(M) =∞, there is a disjoint decomposition⋃∞

j=1 Mj = Mwith µ(Mj) < ∞. The already proven completeness of Lp(Mj, dµ) implies theexistence of functions f (j) ∈ Lp(Mj, dµ) which fulfil ‖f (j)n −f (j)‖p → 0 as n→∞;then we can proceed as in the proof of completeness of �p (cf. Problem 23).

Other examples of complete metric spaces are given in Problem 23. Any metricspace can be extended to become complete: a complete space (X ′, �′) is called thecompletion of (X, �) if (i) X ⊂ X ′ and �′(x, y) = �(x, y) for all x, y ∈ X, and (ii)the set X is everywhere dense in X ′ (this requirement ensures minimality — cf.Problem 25).

1.2.2 Theorem: Any metric space (X, �) has a completion. If (X̃, �̃) is anothercompletion of (X, �), there is an isometry f : X ′ → X̃ which preserves X, i.e.,f(x) = x for all x ∈ X.Sketch of the proof: Uniqueness follows directly from the definition. Existence isproved constructively by the so–called standard completion procedure which genera-lizes the Cantor construction of the reals. We start from the set of all Cauchysequences in (X, �). This can be factorized if we set {xj} ∼ {yj} for the sequenceswith limj→∞ �(xj, yj) = 0. The set of equivalence classes we denote as X

∗ and define�∗([x], [y]) := limj→∞ �(xj, yj) to any [x], [y] ∈ X∗. Finally, one has to check thatthis definition makes sense, i.e., that �∗ does not depend on the choice of sequencesrepresenting the classes [x], [y], �∗ is a metric on X∗, and (X∗, �∗) satisfies therequirements of the definition.

The notion of topology is obtained by axiomatization of some properties ofmetric spaces. Let X be a set and τ a family of its subsets which fulfils the followingconditions (topology axioms):

(t1) X ∈ τ and ∅ ∈ τ .

(t2) If I is any index set and Gα ∈ τ for all α ∈ I; then⋃

α∈I Gα ∈ τ .

(t3)⋂n

j=1 Gj ∈ τ for any finite subsystem {G1, . . . , Gn} ⊂ τ .

The family τ is called a topology, its elements open sets and the set X equippedwith a topology is a topological space; when it is suitable we write (X, τ).

A family of open sets in a metric space (X, �) is a topology by definition;we speak about the metric–induced topology τ�, in particular, the norm–inducedtopology if X is a vector space and � is induced by a norm. On the other hand,finding the conditions under which a given topology is induced by a metric is anontrivial problem (see the notes). Two extreme topologies can be defined on anyset X: the discrete topology τd := 2

X , i.e., the family of all subsets in X, and thetrivial topology τ0 := {∅, X}. The first of them is induced by the discrete metric,�d(x, y) := 1 for x �=y, while (X, τ0) is not metrizable unless X is a one–point set.

An open set in a topological space X containing a point x or a set M ⊂ X iscalled a neighborhood of the point X or the set M , respectively. Using this concept,

8 1 Some notions from functional analysis

we can adapt to topological spaces most of the “metric” definitions presented above,as well as some simple results such as those of Problems 17a, c, 19b, topologicalequivalence of homeomorphic spaces, etc. On the other hand, equally elementarymetric–space properties may not be valid in a general topological space.

1.2.3 Example: Consider the topologies τfin and τcount on X = [0, 1] in whichthe closed sets are all finite and almost countable subsets of X, respectively. If{xn} ⊂ X is a simple sequence, xn �=xm for n �=m ; then any neighborhood U(x)contains all elements of the sequence with the exception of a finite number; hencethe limit is not unique in (X, τfin). This is not the case in (X, τcount) but there onlyvery few sequences converge, namely those with xn = xn0 for all n ≥ n0, whichmeans, in particular, that we cannot use sequences to characterize local continuityor points of the closure.

Some of these difficulties can be solved by introducing a more general notion ofconvergence. A partially ordered set I is called directed if for any α, β ∈ I thereis γ ∈ I such that α ≺ γ and β ≺ γ. A map of a directed index set I into atopological space X, α �→ xα, is called a net in X. A net {xα} is said to convergeto a point x ∈ X if to any neighborhood U(x) there is an α0 ∈ I such thatxα ∈ U(x) for all α � α0. To illustrate that nets in a sense play the role thatsequences played in metric spaces, let us mention two simple results the proofs ofwhich we leave to the reader (Problem 29).

1.2.4 Proposition: Let (X, τ) , (X ′, τ ′) be topological spaces; then

(a) A point x ∈ X belongs to the closure of a set M ⊂ X iff there is a net{xα} ⊂M such that xα → x.

(b) A map f : X → X ′ is continuous at a point x ∈ X iff the net {f(xα)}converges to f(x) for any net {xα} converging to x.

Two topologies can be compared if there is an inclusion between them, τ1 ⊂ τ2,in which case we say that τ1 is weaker (coarser) than τ2; while the latter is stronger(finer) than τ1. Such a relation between topologies has some simple consequences— see, e.g., Problem 32. In particular, continuity of a map f : X → Y is preservedwhen we make the topology in Y weaker or in X stronger. In other cases it maynot be preserved; for instance, Problem 3.9 gives an example of three topologies,τw ⊂ τs ⊂ τu, on a set X := B(H) and a map f : X → X which is continuous withrespect to τw and τu but not τs.

1.2.5 Example: A frequently used way to construct a topology on a given Xemploys a family F of maps from X to a topological space (X̃, τ̃). Among alltopologies such that each f ∈ F is continuous there is one which is the weakest; itsexistence follows from Problem 30, where the system S consists of the sets f (−1)G)for each G ⊂ τ̃ , f ∈ F . We call this the F–weak topology.

1.2 Metric and topological spaces 9

For any set M in a topological space (X, τ) we define the relative topology τMas the family of intersections M ∩ G with G ⊂ τ ; the space (M, τM) is calleda subspace of (X, τ). Other important notions are obtained by axiomatization ofproperties of open balls in metric spaces. A family B ⊂ τ is called a basis of atopological space (X, τ) if any nonempty open set can be expressed as a union ofelements of B. A family Bx of neighborhoods of a given point x ∈ X is called alocal basis at x if any neighborhood U(x) contains some B ∈ Bx. A trivial exampleof both a basis and a local basis is the topology itself; however, we are naturallymore interested in cases where bases are rather a “small part” of it. It is easy to seethat local bases can be used to compare topologies.

1.2.6 Proposition: Let a set X be equipped with topologies τ, τ ′ with local basesBx, B′x at each x ∈ X. The inclusion τ ⊂ τ ′ holds iff for any B ∈ Bx there isB′ ∈ B′x such that B′ ⊂ B.

To be a basis of a topology or a local basis, a family of sets must meet certainconsistency requirements (cf. Problem 30c, d); this is often useful when we define aparticular topology by specifying its basis.

1.2.7 Example: Let (Xj, τj), j = 1, 2, be topological spaces. On the Cartesianproduct X1 ×X2 we define the standard topology τX1×X2 determined by τj, j =1, 2, as the weakest topology which contains all sets G1 × G2 with Gj ∈ τj, i.e.,τX1×X2 := τ(τ1× τ2) in the notation of Problem 30b. Since (A1×A2)∩ (B1×B2) =(A1 ∩ B1) × (A2 ∩ B2), the family τ1 × τ2 itself is a basis of τX1×X2 ; a local basisat [x1, x2] consists of the sets U(x1) × V (x2), where U(x1) ∈ τ1, V (x2) ∈ τ2 areneighborhoods of the points x1, x2, respectively. The space (X1 × X2, τX1×X2) iscalled the topological product of the spaces (Xj, τj), j = 1, 2.

Bases can also be used to classify topological spaces by the so–called countabilityaxioms. A space (X, τ) is called first countable if it has a countable local basis atany point; it is second countable if the whole topology τ has a countable basis. Thesecond requirement is actually stronger; for instance, a nonseparable metric spaceis first but not second countable (cf. Problem 18; some related results are collectedin Problem 31). The most important consequence of the existence of a countablelocal basis, {Un(x) : n = 1, 2, . . .} ⊂ τ , is that one can pass to another local basis{Vn(x) : n = 1, 2, . . .}, which is ordered by inclusion, Vn+1 ⊂ Vn, setting V1 := U1and Vn+1 := Vn ∩ Un+1. This helps to partially rehabilitate sequences as a tool inchecking topological properties (Problem 33a).

The other problem mentioned in Example 1.2.3, namely the possible nonunique-ness of a sequence limit, is not related to the cardinality of the basis but rather to thedegree to which a given topology separates points. It provides another classificationof topological spaces through separability axioms:

T1 To any x, y ∈ X, x �=y, there is a neighborhood U(x) such that y �∈ U(x).

T2 To any x, y ∈ X, x �=y, there are disjoint neighborhoods U(x) and U(y).

10 1 Some notions from functional analysis

T3 To any closed set F ⊂ X and a point x �∈ F , there are disjoint neighborhoodsU(x) and U(F ).

T4 To any pair of disjoint closed sets F, F′, there are disjoint neighborhoods

U(F ) and U(F ′).

A space (X, τ) which fulfils the axioms T1 and Tj is called Tj–space, T2–spacesare also called Hausdorff, T3–spaces are regular, and T4–spaces are normal . Forinstance, the spaces of Example 3 are T1 but not Hausdorff; one can find examplesshowing that the whole hierarchy is nontrivial (see the notes). In particular, anymetric space is normal. The question of limit uniqueness that we started with isanswered affirmatively in Hausdorff spaces (see Problem 29).

1.3 Compactness

One of the central points in an introductory course of analysis is the Heine–Borel the-orem, which claims that given a family of open intervals covering a closed boundedset F ⊂ R, we can select a finite subsystem which also covers F . The notion ofcompactness comes from axiomatization of this result. Let M be a set in a topo-logical space (X, τ). A family P := {Mα : α ∈ I} ⊂ 2X is a covering of M if⋃

α∈I Mα ⊃ M ; in dependence on the cardinality of the index set I the coveringis called finite, countable, etc.We speak about an open covering if P ⊂ τ . The setM is compact if an arbitrary open covering of M has a finite subsystem thatstill covers M ; if this is true for the whole of X we say that the topological space(X, τ) is compact. It is easy to see that compactness of M is equivalent to com-pactness of the space (M, τM) with the induced topology, so it is often sufficient toformulate theorems for compact spaces only.

1.3.1 Proposition: Let (X, τ) be a compact space, then

(a) Any infinite set M ⊂ X has at least one accumulation point.

(b) Any closed set F ⊂ X is compact.

(c) If a map f : (X, τ)→ (X ′, τ ′) is continuous, then f(X) is compact in (X ′, τ ′).

Proof: To check (a) it is obviously sufficient to consider countable sets. SupposeM = {xn : n = 1, 2, . . .} has no accumulation points; then the same is true forthe sets MN := {xn : n ≥ N}. They are therefore closed and their complementsform an open covering of X with no finite subcovering. Further, let {Gα} be anopen covering of F ; adding the set G := X \F we get an open covering of X. Anyfinite subcovering G of X is either contained in {Gα} or it contains the set G;in the latter case G \G is a finite covering of the set F . Finally, the last assertionfollows from the appropriate definitions.

1.3 Compactness 11

Part (a) of the proposition represents a particular case of a more general result(see the notes) which can be used to define compactness; another alternative defini-tion is given in Problem 36. Compactness has an important implication for the wayin which the topology separates points.

1.3.2 Theorem: A compact Hausdorff space is normal.

Proof: Let F, R be disjoint closed sets and y ∈ R. By assumption, to any x ∈ Fone can find disjoint neighborhoods Uy(x) and Ux(y). The family {Uy(x) : x ∈ F}covers the set F , which is compact in view of the previous proposition; hence thereis a finite subsystem {Uy(xj) : j = 1, . . . , n} such that Uy(F ) :=

⋃nj=1 Uy(xj) is a

neighborhood of F . Moreover, U(y) :=⋂n

j=1 Uxj(y) is a neighborhood of the pointy and U(y) ∩ Uy(F ) = ∅. This can be done for any point y ∈ R giving an opencovering {U(y) : y ∈ R} of the set R; from it we select again a finite subsystem{U(yk) : k = 1, . . . , m} such that U(R) :=

⋃mk=1 U(yk) is a neighborhood of R

which has an empty intersection with U(F ) :=⋂m

k=1 Uyk(F ).

1.3.3 Theorem: Let X be a Hausdorff space, then

(a) Any compact set F ⊂ X is closed.

(b) If the space X is compact, then any continuous bijection f : X → X ′ for X ′Hausdorff is a homeomorphism.

Proof: If y �∈ F , the neighborhood U(y) from the preceding proof has an emptyintersection with F , so y �∈ F . To prove (b) we have to check that f(F ) is closed inX ′ for any closed F ⊂ X; this follows easily from (a) and Proposition 1.3.1c.

A set M in a topological space is called precompact (or relatively compact) ifM is compact. A space X is locally compact if any point x ∈ X has a precompactneighborhood; it is σ–compact if any countable covering has a finite subcovering.

Let us now turn to compactness in metric spaces. There, any compact set isclosed by Theorem 1.3.3 and bounded — from an unbounded set we can always selectan infinite subset which has no accumulation point. However, these conditions arenot sufficient. For instance, the closed ball S1(0) in �

2 is bounded but not compact:its subset consisting of the points Xj := {δjk}∞k=1 , j = 1, 2, . . ., has no accumulationpoint because ‖Xj−Xk‖ =

√2 holds for all j �= k.

To be able to characterize compactness by metric properties we need a strongercondition. Given a set M in a metric space (X, �) and ε > 0, we call a set Nεan ε–lattice for M if to any x ∈ M there is a y ∈ Nε such that �(x, y) ≤ ε( Nε may not be a subset of M but by using it one is able to construct a 2ε–lattice for M which is contained in M ). A set M is completely bounded if it has afinite ε–lattice for any ε > 0; if the set X itself is completely bounded we speakabout a completely bounded metric space. If M is completely bounded, the same isobviously true for M . Any completely bounded set is bounded; on the other hand,any infinite orthonormal set in a pre–Hilbert space represents an example of a setwhich is bounded but not completely bounded.

12 1 Some notions from functional analysis

1.3.4 Proposition: A σ–compact metric space is completely bounded. A completelybounded metric space is separable.

Proof: Suppose that for some ε > 0 there is no finite ε–lattice. Then X \Sε(x1) �= ∅for an arbitrarily chosen x1 ∈ X, otherwise {x1} would be an ε–lattice for X. Hencethere is x2 ∈ X such that �(x1, x2) > ε and we have X \(Sε(x1)∪Sε(x2)) �= ∅ etc.;in this way we construct an infinite set {xj : j = 1, 2, . . .} which fulfils �(xj, xk) > εfor all j �= k, and therefore it has no accumulation points. As for the second part,if Nn is a (1/n)–lattice for X, then

⋃∞n=1 Nn is a countable everywhere dense set.

1.3.5 Corollary: Let X be a metric space; then the following conditions are equiv-alent:

(i) X is compact.

(ii) X is σ–compact.

(iii) Any infinite set in X has an accumulation point.

1.3.6 Theorem: A metric space is compact iff it is complete and completelybounded.

Proof: Let X be compact; in view of Proposition 1.3.4 it is sufficient to show thatit is complete. If {xn} is Cauchy, the compactness implies existence of a convergentsubsequence so {xn} is also convergent (Problem 24). On the other hand, to provethe opposite implication we have to check that any M := {xn : n = 1, 2, . . .} ⊂ Xhas an accumulation point. By assumption, there is a finite 1–lattice N1 for X,hence there is y1 ∈ N1 such that the closed ball S1(y1) contains an infinite subsetof M . The ball S1(y1) is completely bounded, so we can find a finite (1/2)–latticeN2 ⊂ S1(y1) and a point y2 ∈ N2 such that the set S1/2(y2) ∩ M is infinite.In this way we get a sequence of closed balls Sn := S21−n(yn) such that each ofthem contains infinitely many points of M and their centers fulfil yn+1 ∈ Sn. Theclosed balls of doubled radii then satisfy S21−n(yn+1) ⊂ S22−n(yn) and M has anaccumulation point in view of Problem 26.

1.3.7 Corollary: (a) A set M in a complete metric space X is precompact iffit is completely bounded. In particular, if X is a finite–dimensional normedspace, then M is precompact iff it is bounded.

(b) A continuous real–valued function f on a compact topological space X isbounded and assumes its maximum and minimum values in X.

Proof: The first assertion follows from Problem 25. If M is compact, it is boundedso M is also bounded. To prove the opposite implication in a finite–dimensionalnormed space, we can use the fact that such a space is topologically isomorphic toC

n (or Rn in the case of a real normed space — see Problem 21). As for part (b),the set f(X) ⊂ R is compact by Proposition 1.3.1c, and therefore bounded. Denote

1.4 Topological vector spaces 13

α := supx∈X f(x) and let {xn} ⊂ X be a sequence such that f(xn)→ α. Since Xis compact there is a subsequence {xkn} converging to some xs and the continuityimplies f(xs) = α. In the same way we can check that f assumes a minimum value.

1.4 Topological vector spaces

We can easily check that the operations of summation and scalar multiplicationin a normed space are continuous. Let us now see what would follow from such arequirement when we combine algebraic and topological properties. A vector spaceV equipped with a topology τ is called a topological vector space if

(tv1) The summation maps continuously (V × V, τV ×V ) to (V, τ).

(tv2) The scalar multiplication maps continuously (C×V, τC×V ) to (V, τ).

(tv3) (V, τ) is Hausdorff.

In the same way, we define a topological vector space over any field. Instead of(tv3), we may demand T1–separability only because the first two requirements implythat T3 is valid (Problem 39).

A useful tool in topological vector spaces is the family of translations,tx : V → V , defined for any x ∈ V by tx(y) := x+ y. Since t−1x = t−x, thecontinuity of summation implies that any translation is a homeomorphism; henceif G is an open set, then x + G := tx(G) is open for all x ∈ V ; in particular, Uis a neighborhood of a point x iff U = x + U(0), where U(0) is a neighborhoodof zero. This allows us to define a topology through its local basis at a single point(Problem 40).

Suppose a map between topological vector spaces (V, τ) and (V ′, τ ′) is simul-taneously an algebraic isomorphism of V, V ′ and a homeomorphism of the corre-sponding topological spaces, then we call it a linear homeomorphism (or topologicalisomorphism). As in the case of normed spaces (cf. Problem 21), the structure of afinite–dimensional topological vector space is fully specified by its dimension.

1.4.1 Theorem: Twofinite–dimensionaltopologicalvectorspaces, (V, τ) and (V ′, τ ′),are linearly homeomorphic iff dim V = dim V ′. Any finite–dimensional topologicalvector space is locally compact.

Proof: It is sufficient to construct a linear homeomorphism of a given n–dimensional(V, τ) to Cn. We take a basis {e1, . . . , en} ⊂ V and construct f : V → Cn byf(∑n

j=1 ξjej

):= [ξ1, . . . , ξn]; in view of the continuity of translations we have to

show that f and f−1 are continuous at zero. According to (tv1), for any U(0) ∈ τwe can find neighborhoods Uj(0) such that

∑nj=1 xj ∈ U(0) for xj ∈ Uj(0) , j =

1, . . . , n and f−1 is continuous by Problem 42a. To prove that f is continuouswe use the fact that V is Hausdorff: Proposition 1.3.1 and Theorem 1.3.3 togetherwith the already proven continuity of f−1 ensure that Sε := {x∈V : ‖f(x)‖ = ε}= f (−1)(Kε) is closed for any ε > 0; we have denoted here by Kε the ε–sphere

14 1 Some notions from functional analysis

in Cn. Since 0 �∈ Sε, the set G := V \ Sε is a neighborhood of zero, and byProblem 42b there is a balanced neighborhood U ⊂ G of zero; this is possible onlyif ‖f(x)‖ < ε for all x ∈ U .

Next we want to discuss a class of topological vector spaces whose properties arecloser to those of normed spaces. In distinction to the latter the topology in them isnot specified generally by a single (semi)norm but rather by a family of them. LetP := {pα : α ∈ I} be a family of seminorms on a vector space V where I is anarbitrary index set. We say that P separates points if to any nonzero x ∈ V thereis a pα ∈ P such that pα(x) �= 0. It is clear that if P consists of a single seminormp it separates points iff p is a norm. Given a family P we set

Bε(p1, . . . , pn) := {x ∈ V : pj(x) < ε , j = 1, . . . , n } ;the collection of these sets for any ε > 0 and all finite subsystems of P will bedenoted as BP0 . In view of Problem 40, BP0 defines a topology on V which wedenote as τP .

1.4.2 Theorem: If a family P of seminorms on a vector space V separates points,then (V, τP) is a topological vector space.

Proof: By assumption, to a pair x, y of different points there is a p ∈ P such thatε := 1

2p(x − y) > 0. Then U(x) := x + Bε(p) and U(y) := y + Bε(p) are disjoint

neighborhoods, so the axiom T2 is valid. The continuity of summation at the point[0, 0] follows from the inequality p(x+y) ≤ p(x)+p(y); for the scalar multiplicationwe use p(αx−α0x0) ≤ |α− α0| p(x0) + |α| p(x−x0).

A topological vector space with a topology induced by a family P separat-ing points is called locally convex. This name has an obvious motivation: if x, y ∈Bε(p1, . . . , pn), then pj(tx+(1− t)x) ≤ tpj(x)+ (1− t)pj(x) holds for any t ∈ [0, 1]so the sets Bε(p1, . . . , pn) are convex. The convexity is preserved at translations, sothe local basis of τP at each x ∈ V consists of convex sets (see also the notes).

1.4.3 Example: The family P := {px := |(x, ·)| : x ∈ V } in a pre–Hilbert space Vgenerates a locally convex topology which is called the weak topology and is denotedas τw; it is easy to see that it is weaker than the “natural” topology induced by thenorm.

1.4.4 Theorem: A locally convex space (V, τ) is metrizable iff there is a countablefamily P of seminorms which generates the topology τ .Proof: If V is metrizable it is first countable. Let {Uj : j = 1, 2, . . .} be a localbasis of τ at the point 0. By definition, to any Uj we can find ε > 0 and afinite subsystem Pj ⊂ P such that

⋂p∈Pj Bε(p) ⊂ Uj. The family P

′ :=⋃∞

j=1Pjis countable and generates a topology τP

′which is not stronger than τ := τP ;

the above inclusion shows that τP′

= τ . On the other hand, suppose that τ isgenerated by a family {pn : n = 1, 2, . . .} separating points; then we can definea metric � as in Problem 16 and show that the corresponding topology satisfiesτ� = τ (Problem 43).

1.5 Banach spaces and operators on them 15

A locally convex space which is complete with respect to the metric used in theproof is called a Fréchet space (see also the notes).

1.4.5 Example: The set S(Rn) consists of all infinitely differentiable functionsf : Rn → C such that

‖f‖J,K := supx∈Rn

|xJ(DKf)(x)|

16 1 Some notions from functional analysis

Proof: Part (a) follows from the appropriate definitions. Suppose that M ={x1, x2, . . .} is total in X and Crat is the countable set of complex numbers with ra-tional real and imaginary parts; then the set L := {

∑nj=1 γjxj : γj ∈ Crat, n 1.

(c) Consider next the space Lp(Rn, dµ) with an arbitrary Borel measure µ onR

n. We use the notation of Appendix A. In particular, J n is the family of allbounded intervals in Rn ; then we define S(n) := {χJ : J ∈ J n}. It is a subsetin Lp and the elements of its linear envelope are called step functions; we cancheck that S(n) is total in Lp(Rn, dµ) (Problem 47). Combining this resultwith Lemma 1.5.2 we see that the subspace C∞0 (R

n) is dense in Lp(Rn, dµ); inparticular, for the Lebesgue measure on Rn the inclusions C∞0 (R

n) ⊂ S(Rn) ⊂Lp(Rn) yield

(C∞0 (Rn))p = (S(Rn))p = Lp(Rn) . (1.2)

(d) Given a topological space (X, τ) we call C∞(X) the set of all continuousfunctions on X with the following property: for any ε > 0 there is a compactset K ⊂ X such that |f(x)| < ε outside K. It is not difficult to check thatC∞(X) is a closed subspace in C(X) and C0(X) = C∞(X), where C0(X)is the set of continuous functions with compact support (Problem 48). In theparticular case X = Rn, C∞0 (R

n) is dense in C∞(Rn) (see the notes), so

(C∞0 (Rn))∞ = (S(Rn))∞ = C∞(Rn) . (1.3)

There are various ways in, which it is possible to construct new Banach spacesfrom given ones. We mention two of them (see also Problem 49):

(i) Let {Xj : j = 1, 2, . . .} be a countable family of Banach spaces. We denoteby X the set of all sequences x := {xj} , xj ∈ Xj, such that

∑j ‖xj‖j

1.5 Banach spaces and operators on them 17

(ii) Starting from the same family {Xj : j = 1, 2, . . .}, one can define anotherBanach space (which is sometimes also referred to as a direct sum) if we changethe above norm to ‖X‖∞ := supj ‖xj‖j replacing, of course, X by the set ofsequences for which ‖X‖∞ < ∞. The two Banach spaces are different unlessthe family {Xj} is finite; the present construction can easily be adapted tofamilies of any cardinality.

A map B : V1 → V2 between two normed spaces is called an operator; inparticular, it is called a linear operator if it is linear. In this case we conventionallydo not use parentheses and write the image of a vector x ∈ V1 as Bx. In this bookwe shall deal almost exclusively with linear operators, and therefore the adjectivewill usually be dropped. A linear operator B : V1 → V2 is said to be bounded ifthere is a positive c such that ‖Bx‖2 ≤ c‖x‖1 for all x ∈ V1; the set of all suchoperators is denoted as B(V1, V2) or simply B(V ) if V1 = V2 := V . One of theelementary properties of linear operators is the equivalence between continuity andboundedness (Problem 50).

The set B(V1, V2) becomes a vector space if we define on it summation andscalar multiplication by (αB + C)x := αBx + Cx. Furthermore, we can associatewith every B ∈ B(V1, V2) the non–negative number

‖B‖ := supS1

‖Bx‖2 ,

where S1 := {x ∈ V1 : ‖x‖1 = 1 } is the unit sphere in V1 (see also Problem 51).

1.5.4 Proposition: The map B �→ ‖B‖ is a norm on B(V1, V2). If V2 is complete,the same is true for B(V1, V2), i.e., it is a Banach space.Proof: The first assertion is elementary. Let {Bn} be a Cauchy sequence in B(V1, V2);then for all n,m large enough we have ‖Bn−Bm‖ < ε, and therefore ‖Bnx−Bmx‖2 ≤ε‖x‖1 for any x ∈ V1. As a Cauchy sequence in V2, {Bnx} converges to someB(x) ∈ V2. The linearity of the operators Bn implies that x �→ Bx is linear,B(x) = Bx. The limit m→∞ in the last inequality gives ‖Bx−Bnx‖2 ≤ ε‖x‖1, soB ∈ B(V1, V2) by the triangle inequality, and ‖B−Bn‖ ≤ ε for all n large enough.

The norm on B(V1, V2) introduced above is called the operator norm. It has anadditional property: if C : V1 → V2 and B : V2 → V3 are bounded operators, andBC is the operator product understood as the composite mapping V1 → V3, we have‖B(Cx)‖3 ≤ ‖B‖ ‖Cx‖2 ≤ ‖B‖ ‖C‖ ‖x‖1 for all x ∈ V1, so BC is also boundedand

‖BC‖ ≤ ‖B‖ ‖C‖ . (1.4)Let V1 be a subspace of a normed space Ṽ1. An operator B : V1 → V2 is called

a restriction of B̃ : Ṽ1 → V2 to the subspace V1 if Bx = B̃x holds for all x ∈ V1,and on the other hand, B̃ is said to be an extension of B; we write B = B̃ |\ V1 orB ⊂ B̃. Another simple property of bounded operators is that they can be extendeduniquely by continuity.

18 1 Some notions from functional analysis

1.5.5 Theorem: Assume that X1, X2 are Banach spaces and V1 is a dense subspacein X1; then any B ∈ B(V1,X2) has just one extension B̃ ∈ B(X1,X2), and moreover,‖B̃‖ = ‖B‖.Proof: For any x ∈ X we can find a sequence {xn} ⊂ V1 that converges to x.Since B is bounded, the sequence {Bxn} is Cauchy, so there is a y ∈ X2 such that‖Bxn−y‖2 → 0. We can readily check that y does not depend on the choice of theapproximating sequence and the map x �→ y is linear; we denote it as B̃. If x ∈ V1one can choose xn = x for all n, which means B̃ |\ V1 = B. Passing to the limit inthe relation ‖Bxn‖2 ≤ ‖B‖ ‖xn‖1 we get B̃ ∈ B(X1,X2) and ‖B̃‖ ≤ ‖B‖; since B̃is an extension of B the two norms must be equal. Suppose finally that B = C |\ V1for some C ∈ B(X1,X2). We have Cxn = Bxn , n = 1, 2, . . ., for any approximatingsequence, and therefore C = B̃.

Notice that in view of Proposition 1.5.4, B(V1,X ) is complete even if V1 is not.The approximation procedure used in the proof to define B̃ is often of practicalimportance, namely if we study an operator whose action on some dense subspaceis given by a simple formula.

1.5.6 Example (Fourier transformation): Following the usual convention we denotethe scalar product of the vectors x, y ∈ Rn by x·y and set

f̂(y) := (2π)−n/2∫

Rn

e−i x·yf(x) dx and f̌(y) := f̂(−y) (1.5)

for any f ∈ S(Rn) and y ∈ Rn. The function f̂ is well–defined and one can checkthat it belongs to S(Rn) (Problem 52), i.e., that F0 : F0f = f̂ is a linear map ofS(Rn) onto itself. We want to prove that F̃0 : f �→ f̌ is its inverse. To this end, weuse the relation from Problem 52; choosing gε := e

−ε2|x|2/2 we get∫

Rn

ei x·y−ε2|x|2/2f̂(y) dy =

∫

Rn

e−|z|2/2f(x+εz) dz

for any ε > 0. The two integrated functions can be majorized independently ofε; the limit ε → 0+ then yields F̃0F0f = f . Since f̌(x) = f̂(−x) we also getF0F̃0f = f for all f ∈ S(Rn), i.e., the relation

(F−10 f)(x) = (F0f)(−x) .Using Theorem 1.5.5, we shall now construct two important extensions of the oper-ator F0. For the moment, we denote by Sp(Rn) the normed space (S(Rn), ‖ · ‖p);we know from (1.2) that it is dense in Lp(Rn) , p ≥ 1.(i) Since S(Rn) is a subset of C∞(Rn) and ‖f̂‖∞ ≤ (2π)−n/2‖f‖1, the operatorF0 can also be understood as an element of B(S1(Rn), C∞(Rn)). As such itextends uniquely to the operator F ∈ B(L1(Rn), C∞(Rn)); it is easy to checkthat its action on any f ∈ L1(Rn) can be expressed again by the first one ofthe relations (1.5),

(Ff)(y) = (2π)−n/2∫

Rn

e−i x·yf(x) dx .

1.5 Banach spaces and operators on them 19

The function Ff is called the Fourier transform of f . We have Ff ∈C∞(R

n), and thereforelim

|x|→∞(Ff)(x) = 0;

this relation is often referred to as the Riemann-Lebesgue lemma.

(ii) Using once more the relation from Problem 52, now with g := f̂ = f̌ , we find∫

Rn

|f̂(y)|2dy =∫

Rn

|f(y)|2dy

for any f ∈ S(Rn). This suggests another possible interpretation of the op-erator F0 as an element of B(S2(Rn), L2(Rn)). Extending it by continuity,we get the operator F ∈ B(L2(Rn)), which is called the Fourier–Planchereloperator or briefly FP–operator; if it is suitable to specify the dimension of Rn

we denote it as Fn. The above relation shows that ‖Ffk‖2 = ‖fk‖2 holds forthe elements of any sequence {fk} ⊂ S2 approximating a given f ∈ L2, andtherefore

‖Ff‖2 = ‖f‖2for all f ∈ L2(Rn); this implies that F is surjective (Problem 53). Hence theFourier–Plancherel operator is a linear isometry of L2(Rn) onto itself.

We are naturally interested in how F acts on the vectors from L2 \ S. Thereis a simple functional realization for n = 1 (see Example 3.1.6). In the generalcase, the right sides of the relations (1.5) express (Ff)(y) and (F−1f)(y) aslong as f ∈ L2 ∩ L1. To check this assume first that supp f ⊂ J , where J isa bounded interval in Rn and consider a sequence {fk} approximating f ac-cording to Problem 47d. By the Hölder inequality ‖f−fk‖1 ≤ µ(J)1/2‖f−fk‖2,so ‖f−fk‖1 → 0, and consequently, the functions F0fk converge uniformly toFf . On the other hand F0fk = Ffk, so ‖F0fk − Ff‖2 = ‖fk−f‖2 → 0; thesought expression of Ff then follows from the result mentioned in the notesto Section 1.2. A similar procedure can be used for a general f ∈ L2 ∩L1: oneapproximates it, e.g., by the functions fj := fχj, where χj are characteristicfunctions of the balls {x ∈ Rn : |x| ≤ j }.For the remaining vectors, f ∈ L2 \L1, the right side of (1.5) no longer makessense and Ff must be defined as a limit, e.g., ‖Ff − Ffj‖2 → 0, where fjare the truncated functions defined above. The last relation is often written as

(Ff)(y) = l.i.m. j→∞(2π)−n/2

∫

|x|≤je−i x·yf(x) dx ,

where the symbol l.i.m. (limes in medio) means convergence with respect tothe norm of L2(Rn).

A particular role is played by operators that map a given normed space intoC. We call B(V, C) the dual space to V and denote it as V ∗ ; its elements are

20 1 Some notions from functional analysis

bounded linear functionals. Comparing it with the algebraic dual space defined inthe notes, we see that V ∗ is a subspace of V f , and moreover, a Banach space withrespect to the operator norm; the two dual spaces do not coincide unless dim V 12 ‖fn‖. In view of Lemma 1.5.2 it is sufficient to checkthat M := {xn : n = 1, 2, . . . } is total in X . Let us assume that V0 := Mlin �= X ;then Proposition 1.5.7 implies the existence of a functional f ∈ X ∗ such that‖f‖ = 1 and f(x) = 0 for x ∈ V0. For any ε > 0 we can find a nonzero fn suchthat ‖fn−f‖ < ε, i.e., ‖fn‖ > 1−ε; hence we arrive at the contradictory conclusionε > ‖fn − f‖ ≥ |fn(xn)− f(xn)| = |fn(xn)| > 12 ‖fn‖ >

12(1− ε).

One of the basic problems in the Banach–space theory is to describe fully X ∗for a given X . We limit ourselves to one example; more information can be found inthe notes and in the next chapter, where we shall show how the problem simplifieswhen X is a Hilbert space.

1.5.9 Example: The dual (�p)∗, p ≥ 1, is linearly isomorphic to �q, where q :=p/(p−1) for p > 1 and q :=∞ for p = 1. To demonstrate this, we define

fY (X) :=∞∑

k=1

ξkηk

1.5 Banach spaces and operators on them 21

for any sequences X := {ξk} ∈ �p and Y := {ηk} ∈ �q; then the Hölder inequalityimplies that fY is a bounded linear functional on �

p and ‖fY ‖ ≤ ‖Y ‖q. The mapY �→ fY of �q to (�p)∗ is obviously linear; we claim that this is the sought isometry.We have to check its invertibility. We take an arbitrary f ∈ (�p)∗ and set ηk :=f(Ek), where Ek are the sequences introduced in Example 1.5.3b; then it followsfrom the continuity of f that the sequence Yf := {ηk}∞k=1 fulfils f = fYf . To showthat Yf ∈ �q, consider first the case p > 1. The vectors Xn :=

∑nk=1 sgn (ηk) |ηk|q−1,

where sgn z := z/|z| if z �= 0 and zero otherwise, fulfil ‖Xn‖p = (∑n

k=1 |ηk|q)1/p

and f(Xn) =∑n

k=1 |ηk|q, so the inequality |f(Xn)| ≤ ‖f‖ ‖Xn‖p yields( ∞∑

k=1

|ηk|q)1/q

≤ ‖f‖ , n = 1, 2, . . . .

If p = 1 we have |f(En)| = |ηn| and ‖En‖1 = 1, so supn |ηn| ≤ ‖f‖. Hence in bothcases the sequence Yf ∈ �q, and the obtained bounds to its norm in combinationwith the inequality ‖fY ‖ ≤ ‖Y ‖q, which we proved above, yield ‖f‖ = ‖Yf‖q.

The dual space of a given V is normed, so we can define the second dual V ∗∗ :=(V ∗)∗ as well as higher dual spaces. For any x ∈ V we can define Jx ∈ V ∗ byJx(f) := f(x). The map x �→ Jx is a linear isometry of V to a subspace of V ∗∗(Problem 55); if its range is the whole V ∗∗ the space V is called reflexive. It followsfrom the definition that any reflexive space is automatically Banach. In view ofExample 1.5.9, the spaces �p are reflexive for p > 1, and the same is true forLp(M, dµ) (see the notes). On the other hand, �1 and C(K) are not reflexive,and similarly, L1(M, dµ) is not reflexive unless the measure µ has a finite support.Below we shall need the following general property of reflexive spaces, which wepresent without proof.

1.5.10 Theorem: Any closed subspace of a reflexive space is reflexive.

The notion of dual space extends naturally to topological vector spaces: thedual to (V, τ) consists of all continuous linear functionals on V ; in this case weoften denote it alternatively as V ′. It allows us to define the weak topology τw onV as the weakest topology with respect to which any f ∈ V ′ is continuous, orthe V ′–weak topology in the terminology introduced in Example 1.2.5 ; in the nextchapter we shall see that the definition is consistent with that of Example 1.4.3. Wehave τw ⊂ τ because each f ∈ V ′ is by definition continuous with respect to τ ,and it is easy to check that τw coincides with the topology generated by the familyPw := { pf : f ∈ V ′ }, where pf (x) := |f(x)|.

1.5.11 Proposition: If (V, τ) is a locally convex space, then Pw is separatingpoints of V and the space (V, τw) is also locally convex.

Proof: If V is a normed space, the assertion follows from Corollary 1.5.8; for thegeneral case see the notes.

If a sequence {xn} ⊂ V converges to some x ∈ V with respect to τw, we write

22 1 Some notions from functional analysis

x = w limn→∞xn or xnw→ x. Let us list some properties of weakly convergent

sequences (only the case dim V =∞ is nontrivial — see Problem 56).

1.5.12 Theorem: (a) xnw→ x iff f(xn)→ f(x) holds for any f ∈ V ∗.

(b) Any weakly convergent sequence in a normed space is bounded.

(c) If {xn} is a bounded sequence in a normed space V and g(xn)→ g(x) for allg of some total set in F ⊂ V ∗, then xn w→ x.

Proof: The first assertion follows directly from the definition. Further, we use theuniform boundedness principle which will be proven in the next section: if xn

w→x, then the family {ψn} ⊂ V ∗∗ with ψn(f) := f(xn) fulfils the assumptions ofTheorem 1.6.1, which yields ‖ψn‖ = ‖xn‖ < c for some c > 0. As for part (c), wehave g(xn) → g(x) for all g ∈ Flin. Since F is total by assumption, for any f ∈V ∗ , ε > 0 there is a g ∈ Flin and a positive integer n(ε) such that ‖f−g‖ < ε and|g(xn)−g(x)| < ε for n > n(ε); this easily yields |f(xn)−f(x)| ≤ (1+‖xn‖+‖x‖)ε.However, the sequence {xn} is bounded, so f(xn)→ f(x).

1.5.13 Example (weak convergence in �p, p > 1 ): In view of Examples 3b and9, the family of the functionals fk({ξj}∞j=1) := ξk , k = 1, 2, . . ., is total in (�p)∗.This means that a sequence {Xn} ⊂ �p, Xn := {ξ(n)j }∞j=1, converges weakly toX := {ξj}∞j=1 ∈ �p iff it is bounded and ξ

(n)j → ξj for j = 1, 2, . . . . For instance,

the sequence {En} of Example 3b converges weakly to zero; this illustrates that thetwo topologies are different because {En} is not norm–convergent.

A topological vector space (V, τ) is called weakly complete if any sequence{xn} ⊂ V such that {f(xn)} is convergent for each f ∈ V ′ converges weaklyto some x ∈ V . A set M ⊂ V is weakly compact if any sequence {xn} ⊂ Mcontains a weakly convergent subsequence.

1.5.14 Theorem: Let X be a reflexive Banach space, then

(a) X is weakly complete.

(b) A set M ⊂ X is weakly compact iff it is bounded.

Proof: (a) Let {xn} ⊂ X be such that {f(xn)} is convergent for each f ∈ V ∗. Thesame argument as in the proof of Theorem 12 implies existence of a positive c suchthat the sequence {ψn} ⊂ X ∗∗, ψn(f) := f(xn), fulfils |ψn(f)| ≤ c‖f‖ , n = 1, 2, . . .,for all f ∈ X ∗. The limit ψ(f) := limn→∞ ψn(x) exists by assumption, the mapf �→ ψ(f) is linear, and the last inequality implies f ∈ X ∗∗. Since X is reflexive,there is an x ∈ X such that ψ(f) = f(x) for all f ∈ X ∗, i.e., xn w→ x.

(b) If M is not bounded there is a sequence {xn} ⊂ M such that ‖xn‖ > n;then no subsequence of it can be weakly convergent. Suppose on the contrary thatM is bounded and consider a sequence X := {xn} ⊂ M ; it is clearly sufficient to

1.6 The principle of uniform boundedness 23

assume that X is simple, xn �= xm for n �= m. In view of Theorem 10, Y := {xn}linis a separable and reflexive Banach space, so Y∗∗ is also separable, and Corollary 8implies that Y∗ is separable too. Let {gj : j = 1, 2, . . . } be a dense set in Y∗.Since X is bounded the same is true for {g(xn)}; hence there is a subsequenceX1 := {x(1)n } such that {g1(x(1)n )} converges. In a similar way, {g2(x(1)n )} is bounded,so we can pick a subsequence X2 := {x(2)n } ⊂ X1 such that {g2(x(2)n )} converges,etc.This procedure yields a chain of sequences, X ⊃ · · · ⊃ Xj ⊃ Xj+1 ⊃ · · ·, suchthat {gj(x(j)n )}∞n=1 , j = 1, 2, . . ., are convergent. Now we set yn := x

(n)n , so yn ∈ Xj

for n ≥ j and {gj(yn)}∞n=1 converges for any j; then {g(yn)}∞n=1 is convergent forall g ∈ Y∗ due to Theorem 12c. The already proven part (a) implies the existence ofy ∈ Y such that g(yn)→ g(y) for any g ∈ Y∗. Finally, we take an arbitrary f ∈ X ∗and denote gf := f |\ Y . Since gf ∈ Y∗ we have f(yn) = gf (yn) → gf (y) = f(y),and therefore yn

w→ y.

1.6 The principle of uniform boundedness

Any Banach space X is a complete metric space so the Baire category theoremis valid in it (cf. Problem 27). Now we are going to use this fact to derive someimportant consequences for bounded operators on X .

1.6.1 Theorem (uniform boundedness principle): Let F ⊂ B(X , V1), where X isa Banach space and (V1, ‖ · ‖1) is a normed space. If supB∈F ‖Bx‖1 < ∞ for anyx ∈ X , then there is a positive c such that supB∈F ‖B‖ < c.Proof: Since any operator B ∈ F is continuous, the sets Mn := {x ∈ X :‖Bx‖1 ≤ n for all B ∈ F } are closed. Due to the assumption, we have X =⋃∞

n=1 Mn and by the Baire theorem, at least one of the sets Mn has an interiorpoint, i.e., there is a natural number ñ, an x̃ ∈Mñ, and an ε > 0 such that all xfulfilling ‖x−x̃‖ < ε belong to Mñ, and therefore supB∈F ‖Bx‖1 ≤ ñ. Let y ∈ Xbe a unit vector. We set xy :=

ε2y; then xy+x̃ ∈Mñ and

‖By‖1 =2

ε‖Bxy‖1 ≤

2

ε(‖B(xy+x̃)‖1 + ‖Bx̃‖1) ≤

4ñ

ε;

this implies ‖B‖ ≤ 4ñε

for all B ∈ F .

In what follows, X , Y are Banach spaces, Uε and Vε are open balls in X andY , respectively, of the radius ε > 0 centered at zero. By N o we denote the interiorof a set N ⊂ Y , i.e., the set of all its interior points. Any operator B ∈ B(X ,Y) iscontinuous, so the pull–back B(−1)(G) of an open set G ⊂ Y is open in X . If Bis surjective, the converse is also true.

1.6.2 Theorem (open–mapping theorem): If an operator B ∈ B(X ,Y) is surjectiveand G ⊂ X is an open set, then the set BG is open in Y .

We shall first prove a technical result.

24 1 Some notions from functional analysis

1.6.3 Lemma: Let B ∈ B(X ,Y) and ε > 0. If (BUε)o �= ∅ or (BUε)o �= ∅; then0 ∈ (BUη)0 or 0 ∈ (BUη)0, respectively, holds for any η > 0.Proof: Let y0 be an interior point of BUε; then there is δ > 0 such that y0 +Vδ ⊂ BUε, i.e., to any y ∈ Vδ there exists a sequence {x(y)n } ⊂ Uε such thatz

(y)n := Bx

(y)n → y + y0. In particular, z(0)n → y0, so z(y)n − z(0)n → y, and since

‖x(y)n −x(o)n ‖X < 2ε we get Vδ ⊂ BU2ε. In view of Problem 33, cBUε = cBUε holdsfor any c > 0. This implies that Vη′ ⊂ BUη , η′ := ηδ2ε , so 0 is an interior point ofBUη. A similar argument applies to (BUε)

o �= ∅.

Proof of Theorem 1.6.2: To any x ∈ G we can find Uη such that x + Uη ⊂ G, i.e.,Bx+BUη ⊂ BG. If there is Vδ ⊂ BUη the set BG is open; hence it is sufficient tocheck 0 ∈ (BUη)o for any η > 0. We write X =

⋃∞n=1 Un; since B is surjective, we

have Y =⋃∞

n=1 BUn and the Baire category theorem implies (BUñ)o �= ∅ for some

positive integer ñ. We shall prove that BUñ ⊂ BU2ñ.Due to the lemma, BUñ contains a ball Vδ, and this further implies Vδj ⊂

BUnj for j = 1, 2, . . ., where δj := δ/2j and nj := ñ/2

j. Let y ∈ BUñ, so anyneighborhood of y contains elements of BUñ; in particular, for the neighborhoody+Vδ1 we can find x1 ∈ Uñ such that Bx1 ∈ y+Vδ1 , and therefore also y−Bx1 ∈Vδ1 ⊂ BUn1 . Repeating the argument we see that there is an x2 ∈ Un1 such thaty−Bx1−Bx2 ∈ Vδ2 ⊂ BUn2 etc; in this way we construct a sequence {xj} ⊂ Xsuch that ‖xj‖X < 2ñ/2j and

∥∥∥ y −

j∑

k=1

Bxk

∥∥∥Y

< δj .

Then Theorem 1.6.1 implies the existence of limj→∞∑j

k=1 xj =: x ∈ X ; we havey = Bx because B is continuous. Now ‖x‖X ≤

∑∞k=1 ‖xk‖X < 2ñ, so y ∈ BU2ñ;

this proves BUñ ⊂ BU2ñ. Since (BUñ)o �= ∅ the set BU2ñ has an interior point;using the lemma again we find 0 ∈ (BUη)o for any η > 0.

Theorem 1.6.2 further implies the following often used result, the proof of whichis left to the reader (Problem 58).

1.6.4 Corollary (inverse–mapping theorem): If B ∈ B(X ,Y) is a bijection, thenB−1 is a continuous linear operator