GRADUATE STUDIES IN MATHEMATICS 191

Functional Analysis

Theo Bühler Dietmar A. Salamon

Functional Analysis

10.1090/gsm/191

GRADUATE STUDIES IN MATHEMATICS 191

Functional Analysis

Theo Bühler Dietmar A. Salamon

EDITORIAL COMMITTEE

Dan AbramovichDaniel S. Freed (Chair)

Gigliola StaffilaniJeff A. Viaclovsky

2010 Mathematics Subject Classification. Primary 46-01, 47-01.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-191

Library of Congress Cataloging-in-Publication Data

Names: Buhler, Theo, 1978– author. | Salamon, D. (Dietmar), author.Title: Functional analysis / Theo Buhler, Dietmar A. Salamon.Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Gradu-

ate studies in mathematics ; volume 191 | Includes bibliographical references and index.Identifiers: LCCN 2017057159 | ISBN 9781470441906 (alk. paper)Subjects: LCSH: Functional analysis. | Spectral theory (Mathematics) | Semigroups of operators.

| AMS: Functional analysis – Instructional exposition (textbooks, tutorial papers, etc.). msc |Operator theory – Instructional exposition (textbooks, tutorial papers, etc.). msc

Classification: LCC QA320 .B84 2018 | DDC 515/.7—dc23LC record available at https://lccn.loc.gov/2017057159

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10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18

Contents

Preface ix

Introduction xi

Chapter 1. Foundations 1

§1.1. Metric Spaces and Compact Sets 2

§1.2. Finite-Dimensional Banach Spaces 17

§1.3. The Dual Space 25

§1.4. Hilbert Spaces 31

§1.5. Banach Algebras 35

§1.6. The Baire Category Theorem 40

§1.7. Problems 45

Chapter 2. Principles of Functional Analysis 49

§2.1. Uniform Boundedness 50

§2.2. Open Mappings and Closed Graphs 54

§2.3. Hahn–Banach and Convexity 65

§2.4. Reflexive Banach Spaces 80

§2.5. Problems 101

Chapter 3. The Weak and Weak* Topologies 109

§3.1. Topological Vector Spaces 110

§3.2. The Banach–Alaoglu Theorem 124

§3.3. The Banach–Dieudonne Theorem 130

§3.4. The Eberlein–Smulyan Theorem 134

v

vi Contents

§3.5. The Kreın–Milman Theorem 140

§3.6. Ergodic Theory 144

§3.7. Problems 153

Chapter 4. Fredholm Theory 163

§4.1. The Dual Operator 164

§4.2. Compact Operators 173

§4.3. Fredholm Operators 179

§4.4. Composition and Stability 184

§4.5. Problems 189

Chapter 5. Spectral Theory 197

§5.1. Complex Banach Spaces 198

§5.2. Spectrum 208

§5.3. Operators on Hilbert Spaces 222

§5.4. Functional Calculus for Self-Adjoint Operators 234

§5.5. Gelfand Spectrum and Normal Operators 246

§5.6. Spectral Measures 261

§5.7. Cyclic Vectors 281

§5.8. Problems 288

Chapter 6. Unbounded Operators 295

§6.1. Unbounded Operators on Banach Spaces 295

§6.2. The Dual of an Unbounded Operator 306

§6.3. Unbounded Operators on Hilbert Spaces 313

§6.4. Functional Calculus and Spectral Measures 326

§6.5. Problems 342

Chapter 7. Semigroups of Operators 349

§7.1. Strongly Continuous Semigroups 350

§7.2. The Hille–Yosida–Phillips Theorem 363

§7.3. The Dual Semigroup 377

§7.4. Analytic Semigroups 388

§7.5. Banach Space Valued Measurable Functions 404

§7.6. Inhomogeneous Equations 425

§7.7. Problems 439

Contents vii

Appendix A. Zorn and Tychonoff 445

§A.1. The Lemma of Zorn 445

§A.2. Tychonoff’s Theorem 449

Bibliography 453

Notation 459

Index 461

Preface

These are notes for the lecture course “Functional Analysis I” held by thesecond author at ETH Zurich in the fall semester 2015. Prerequisites arethe first year courses on Analysis and Linear Algebra, and the second yearcourses on Complex Analysis, Topology, and Measure and Integration.

The material of Section 1.4 on elementary Hilbert space theory, Sub-section 5.4.2 on the Stone–Weierstraß Theorem, and the appendix on theLemma of Zorn and Tychonoff’s Theorem was not covered in the lectures.These topics were assumed to have been covered in previous lecture courses.They are included here for completeness of the exposition.

The material of Subsection 2.4.4 on the James space, Section 5.5 on thefunctional calculus for bounded normal operators, and Chapter 6 on un-bounded operators was not part of the lecture course (with the exception ofsome of the basic definitions in Chapter 6 that are relevant for infinitesimalgenerators of strongly continuous semigroups). From Chapter 7 only thebasic material on strongly continuous semigroups in Section 7.1, on theirinfinitesimal generators in Section 7.2, and on the dual semigroup in Sec-tion 7.3 were included in the lecture course.

23 October 2017 Theo Buhler

Dietmar A. Salamon

ix

Introduction

Classically, functional analysis is the study of function spaces and linearoperators between them. The relevant function spaces are often equippedwith the structure of a Banach space and many of the central results re-main valid in the more general setting of bounded linear operators betweenBanach spaces or normed vector spaces, where the specific properties ofthe concrete function space in question only play a minor role. Thus, in themodern guise, functional analysis is the study of Banach spaces and boundedlinear operators between them, and this is the viewpoint taken in the presentbook. This area of mathematics has both an intrinsic beauty, which we hopeto convey to the reader, and a vast number of applications in many fields ofmathematics. These include the analysis of PDEs, differential topology andgeometry, symplectic topology, quantum mechanics, probability theory, geo-metric group theory, dynamical systems, ergodic theory, and approximationtheory, among many others. While we say little about specific applications,they do motivate the choice of topics covered in this book, and our goal isto give a self-contained exposition of the necessary background in abstractfunctional analysis for many of the relevant applications.

The book is addressed primarily to third year students of mathematicsor physics, and the reader is assumed to be familiar with first year analysisand linear algebra, as well as complex analysis and the basics of point settopology and measure and integration. For example, this book does notinclude a proof of completeness and duality for Lp spaces.

There are naturally many topics that go beyond the scope of the presentbook, such as Sobolev spaces and PDEs, which would require a book on itsown and, in fact, very many books have been written about this subject;here we just refer the interested reader to [19,28,30]. We also restrict the

xi

xii Introduction

discussion to linear operators and say nothing about nonlinear functionalanalysis. Other topics not covered include the Fourier transform (see [2,48,79]), maximal regularity for semigroups (see [76]), the space of Fredholmoperators on an infinite-dimensional Hilbert space as a classifying space forK-theory (see [5–7,42]), Quillen’s determinant line bundle over the space ofFredholm operators (see [71,77]), and the work of Gowers [31] and Argyros–Haydon [4] on Banach spaces on which every bounded linear operator is thesum of a scalar multiple of the identity and a compact operator. Here is adescription of the contents of the book, chapter by chapter.

Chapter 1 discusses some basic concepts that play a central role in thesubject. It begins with a section on metric spaces and compact sets whichincludes a proof of the Arzela–Ascoli Theorem. It then moves on to establishsome basic properties of finite-dimensional normed vector spaces and shows,in particular, that a normed vector space is finite-dimensional if and only ifthe unit ball is compact. The first chapter also introduces the dual space of anormed vector space, explains several important examples, and contains anintroduction to elementary Hilbert space theory. It then introduces Banachalgebras and shows that the group of invertible elements is an open set. Itcloses with a proof of the Baire Category Theorem.

Chapter 2 is devoted to the three fundamental principles of functionalanalysis. They are theUniform Boundedness Principle (a pointwise boundedfamily of bounded linear operators on a Banach space is bounded), the OpenMapping Theorem (a surjective bounded linear operator between Banachspaces is open), and the Hahn–Banach Theorem (a bounded linear func-tional on a linear subspace of a normed vector space extends to a boundedlinear functional on the entire normed vector space). An equivalent formu-lation of the Open Mapping Theorem is the Closed Graph Theorem (a linearoperator between Banach spaces is bounded if and only if it has a closedgraph) and a corollary is the Inverse Operator Theorem (a bijective boundedlinear operator between Banach spaces has a bounded inverse). A slightlystronger version of the Hahn–Banach Theorem, with the norm replaced bya quasi-seminorm, can be reformulated as the geometric assertion that twoconvex subsets of a normed vector space can be separated by a hyperplanewhenever one of them has nonempty interior. The chapter also discussesreflexive Banach spaces and includes an exposition of the James space.

The subjects of Chapter 3 are the weak topology on a Banach space Xand the weak* topology on its dual space X∗. With these topologies Xand X∗ are locally convex Hausdorff topological vector spaces and the chap-ter begins with a discussion of the elementary properties of such spaces. Thecentral result of the third chapter is the Banach–Alaoglu Theorem whichasserts that the unit ball in the dual space is compact with respect to the

Introduction xiii

weak* topology. This theorem has important consequences in many fields ofmathematics. The chapter also contains a proof of the Banach–DieudonneTheorem which asserts that a linear subspace of the dual space of a Banachspace is weak* closed if and only if its intersection with the closed unit ballis weak* closed. A consequence of the Banach–Alaoglu Theorem is that theunit ball in a reflexive Banach space is weakly compact, and the Eberlein–Smulyan Theorem asserts that this property characterizes reflexive Banachspaces. The Kreın–Milman Theorem asserts that every nonempty compactconvex subset of a locally convex Hausdorff topological vector space is theclosed convex hull of its extremal points. Combining this with the Banach–Alaoglu Theorem, one can prove that every homeomorphism of a compactmetric space admits an invariant ergodic Borel probability measure. Someproperties of such ergodic measures can be derived from an abstract func-tional analytic ergodic theorem which is also established in this chapter.

The purpose of Chapter 4 is to give a basic introduction to Fredholmoperators and their indices including the stability theorem. A Fredholmoperator is a bounded linear operator between Banach spaces that has afinite-dimensional kernel, a closed image, and a finite-dimensional cokernel.Its Fredholm index is the difference of the dimensions of kernel and cokernel.The stability theorem asserts that the Fredholm operators of any given indexform an open subset of the space of all bounded linear operators between twoBanach spaces, with respect to the topology induced by the operator norm.It also asserts that the sum of a Fredholm operator and a compact operator isagain Fredholm and has the same index as the original operator. The chapterincludes an introduction to the dual of a bounded linear operator, a proof ofthe Closed Image Theorem which asserts that an operator has a closed imageif and only if its dual does, an introduction to compact operators which mapthe unit ball to pre-compact subsets of the target space, a characterizationof Fredholm operators in terms of invertibility modulo compact operators,and a proof of the stability theorem for Fredholm operators.

The purpose of Chapter 5 is to study the spectrum of a bounded linearoperator on a real or complex Banach space. A first preparatory sectiondiscusses complex Banach spaces and the complexifications of real Banachspaces, the integrals of continuous Banach space valued functions on com-pact intervals, and holomorphic operator valued functions. The chapter thenintroduces the spectrum of a bounded linear operator, examines its elemen-tary properties, discusses the spectra of compact operators, and establishesthe holomorphic functional calculus. The remainder of this chapter dealsexclusively with operators on Hilbert spaces, starting with a discussion ofcomplex Hilbert spaces and the spectra of normal and self-adjoint operators.It then moves on to C* algebras and the continuous functional calculus forself-adjoint operators, which takes the form of an isomorphism from the

xiv Introduction

C* algebra of complex valued continuous functions on the spectrum to thesmallest C* algebra containing the given operator. The next topic is theGelfand representation and the extension of the continuous functional cal-culus to normal operators. The chapter also contains a proof that everynormal operator can be represented by a projection valued measure on thespectrum, and that every self-adjoint operator is isomorphic to a direct sumof multiplication operators on L2 spaces.

Chapter 6 is devoted to unbounded operators and their spectral the-ory. The domain of an unbounded operator on a Banach space is a linearsubspace. In most of the relevant examples the domain is dense and the op-erator has a closed graph. The chapter includes a discussion of the dual of anunbounded operator and an extension of the Closed Image Theorem to thissetting. It then examines the basic properties of the spectra of unboundedoperators. The remainder of the chapter deals with unbounded operatorson Hilbert spaces and their adjoints. In particular, it extends the functionalcalculus and the spectral measure to unbounded self-adjoint operators.

Strongly continuous semigroups of operators are the subject of Chap-ter 7. They play an important role in the study of many linear partialdifferential equations such as the heat equation, the wave equation, and theSchrodinger equation, and they can be viewed as infinite-dimensional ana-logues of the exponential matrix S(t) := etA. In all the relevant examplesthe operator A is unbounded. It is called the infinitesimal generator of thestrongly continuous semigroup in question. A central result in the subjectis the Hille–Yosida–Phillips Theorem which characterizes the infinitesimalgenerators of strongly continuous semigroups. The dual semigroup is notalways strongly continuous. It is, however, strongly continuous wheneverthe Banach space in question is reflexive. The chapter also includes a basictreatment of analytic semigroups and their infinitesimal generators. It closeswith a study of Banach space valued measurable functions and of solutionsto the inhomogeneous equation associated to a semigroup.

Each of the seven chapters ends with a problem section, which we hopewill give the interested reader the opportunity to deepen their understandingof the subject.

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Notation

2X , set of all subsets of a set X, 2

A, Banach algebra, 35

A∗, dual operator, 164 (real), 198 (complex), 306 (unbounded)

A∗, adjoint operator, 165 (real), 225 (complex), 313 (unbounded)

A, space of unital algebra homomorphism Λ : A → C, 249, 293

B ⊂ 2M , Borel σ-algebra, 30

c0, space of sequences converging to zero, 29

C(M), space of continuous functions, 30

C(X,Y ), space of continuous maps, 12

coker(A), cokernel of an operator, 179

conv(S), convex hull, 120

conv(S), closed convex hull, 120

Δ, Laplace operator, 298, 320

dom(A), domain of an unbounded operator, 59, 295

graph(A), graph of an operator, 59

G ⊂ A, group of invertible elements in a unital Banach algebra, 35

H, Hilbert space, 31 (real), 223 (complex)

im(A), image of an operator, 18, 179

index(A), Fredholm index, 179

ker(A), kernel of an operator, 18, 179

L(X,Y ), space of bounded linear operators, 17

L(X) = L(X,X), space of bounded linear endomorphisms, 36

459

460 Notation

Lc(X,Y ), space of bounded complex linear operators, 198

Lc(X) = Lc(X,X), space of bounded complex linear endomorphisms, 198

�p, space of p-summable sequences, 3

�∞, space of bounded sequences, 3

Lp(μ) = Lp(μ)/∼, Banach space of p-integrable functions, 4

L∞(μ) = L∞(μ)/∼, Banach space of bounded measurable functions, 4

Lp(I,X), Banach space of Banach space valued p-integrable functions, 409

(M,A, μ), measure space, 3

M(M,A), space of signed measures, 4

M(M), space of signed Borel measures, 30

M(φ), set of φ-invariant Borel probability measures, 125

ρ(A), resolvent set, 208 (bounded), 299 (unbounded)

Rλ(A) = (λ1l−A)−1, resolvent operator, 210 (bounded), 299 (unbounded)

σ(A), spectrum of an operator, 208 (bounded), 299 (unbounded)

Spec(A), spectrum of a commutative unital Banach algebra, 246

S(t), strongly continuous semigroup, 350

S⊥, orthogonal complement, 225

S⊥, annihilator, 74⊥T , pre-annihilator, 120

U (X, d), topology of a metric space, 2

U (X, ‖·‖), topology of a normed vector space, 3

V ⊂ H ⊂ V ∗, Gelfand triple, 316

W 1,1(I), Banach space of absolutely continuous functions, 194,

W 1,p(I,C), Sobolev space on an interval I ⊂ R, 194, 297, 304, 315, 353

W 1,∞(R,C), Banach space of Lipschitz continuous functions, 297,

W 2,p(Rn,C), Sobolev space on Rn, 298, 386 (p = 2, n = 1)

W 2,2(Ω) ∩W 1,20 (Ω), Sobolev space and Dirichlet problem on Ω ⊂ Rn, 320

W 1,p(I,X), Sobolev space of Banach space valued functions, 423

Xc, complexification of a real vector space, 201

(X, d), metric space, 2

(X, ‖·‖), normed vector space, 3 (real), 198 (complex)

X∗ = L(X,R), dual space, 26

X∗ = Lc(X,C), complex dual space, 198

X∗∗ = L(X∗,R), bidual space, 80

〈x∗, x〉, pairing of a normed vector space with its dual space, 74

Index

absolutely continuous, 413

adjoint operator, 165

complex, 225

unbounded, 313

affine hyperplane, 72

Alaoglu–Bourbaki Theorem, 156

almost everywhere, 4annihilator, 74

left, 120

approximation property, 178

Argyros–Haydon space, 188

Arzela–Ascoli Theorem, 13, 15, 16, 155

Atiyah–Janich Theorem, 188

axiom of choice, 446

Bell–Fremlin Theorem, 162

axiom of countable choice, 6

axiom of dependent choice, 6, 48

Babylonian method

for square roots, 236

Baire Category Theorem, 42

Banach algebra, 35, 234

ideal, 246semisimple, 246

Banach hyperplane problem, 188

Banach limit, 104

Banach space, 3

approximation property, 178

complex, 100, 198

complexified, 201

product, 25

quotient, 24

Radon–Nikodym property, 415

reflexive, 81, 134

separable, 85

strictly convex, 143

Banach–Alaoglu Theorem

general case, 126, 155

separable case, 124

Banach–Dieudonne Theorem, 130Banach–Mazur Theorem, 158

Banach–Steinhaus Theorem, 52

basis

orthonormal, 79

Schauder, 99, 106, 178

Bell–Fremlin Theorem, 162

bidual

operator, 165

space, 80

bilinear form

continuous, 53

positive definite, 31symmetric, 31

symplectic, 314, 345

Birkhoff’s Ergodic Theorem, 145

Birkhoff–von Neumann Theorem, 160

Borel σ-algebra, 30

Borel measurable operator, 48

bounded

bilinear map, 53

linear operator, 17

invertible, 39

pointwise, 50Bourbaki–Witt Theorem, 446

C* algebra, 234

461

462 Index

Calkin algebra, 187Cantor function, 158category

in the sense of Baire, 40Cauchy integral formula, 207Cauchy problem, 349, 363

well-posed, 363Cauchy sequence, 2Cauchy–Schwarz inequality, 31

complex, 222Cayley transform, 327Cayley–Hamilton Theorem, 291chain, 445closeable linear operator, 62closed convex hull, 120Closed Graph Theorem, 59Closed Image Theorem, 169, 308closed linear operator, 59cokernel, 179comeagre, 40compact

subset of a topological space, 5finite intersection property, 450operator, 173–177pointwise, 12subset of a metric space, 5

compact-open topology, 154complemented subspace, 78complete

metric space, 3subset of a metric space, 5

completely continuous operator, 174completion of a metric space, 45complexification

of a linear operator, 201of a norm, 201of a vector space, 201of the dual space, 202

continuous functionvanishing at infinity, 129, 155weakly, 404

contraction semigroup, 374convergence

in measure, 111weak, 114weak*, 114

convex hull, 120convex set

absorbing, 104closure and interior, 73, 115extremal point, 140

face, 140separation, 70, 116, 123

cyclic vector, 281

deformation retract, 193dense

linear subspace, 76subset, 11

direct sum, 57Dirichlet problem, 320dissipative operator, 374doubly stochastic matrix, 160dual operator, 164, 306

complex, 198dual space, 26

complex, 198of �1, 29of �p, 28of C(M), 30of c0, 29of Lp(μ), 26of a Hilbert space, 32of a quotient, 76of a subspace, 76

Dunford integral, 216, 305

Eberlein–Smulyan Theorem, 134eigenspace

generalized, 213eigenvalue, 208, 299eigenvector, 208, 299equi-continuous, 12, 16equivalent norms, 18ergodic

measure, 144theorem, 148

Birkhoff, 145von Neumann, 146

uniquely, 145exact sequence, 195

Euler characteristic, 195extremal point, 140

Fejer’s Theorem, 79finite intersection property, 450first category, 40flow, 352formal adjoint

of a differential operator, 64Fourier series, 79, 103Fredholm

alternative, 187, 193

Index 463

index, 179operator, 179Stability Theorem, 185triple, 194

functional calculusbounded measurable, 267continuous, 240, 257holomorphic, 217, 305normal, 257, 267self-adjoint, 240, 326unbounded, 326

Gantmacher’s Theorem, 190Gelfand representation, 249Gelfand spectrum, 246, 293Gelfand transform, 249, 293Gelfand triple, 316Gelfand–Mazur Theorem, 248, 290Gelfand–Robbin quotient, 315, 346graph norm, 59, 296, 361

Hahn–Banach Theorem, 65closure of a subspace, 74for bounded linear functionals, 67for convex sets, 70, 116, 123for positive linear functionals, 68

Hardy space, 239heat

equation, 352, 403kernel, 352, 403

Hellinger–Toeplitz Theorem, 61Helly’s Theorem, 135, 158Hermitian inner product, 222

on �2(N,C), 224on L2(μ,C), 224on L2(R/Z,C), 79

Hilbert cube, 143, 161Hilbert space, 31

complex, 223complexification, 223dual space, 26orthonormal basis, 79separable, 79unit sphere contractible, 193

Hille–Yosida–Phillips Theorem, 368–374Holder inequality, 26holomorphic

function, 205functional calculus, 216–221

hyperplane, 72affine, 72

image, 18infinitesimal generator, 357

of a contraction semigroup, 375of a group, 366of a self-adjoint semigroup, 382of a shift group, 385of a unitary group, 384of an analytic semigroup, 393of the dual semigroup, 377of the heat semigroup, 403Schrodinger operator, 386uniqueness of the semigroup, 365well-posed Cauchy problem, 363

inner product, 31Hermitian, 79, 222on L2(μ), 34

integralBanach space valued, 203, 410mean value inequality, 204over a curve, 205

invariant measure, 125ergodic, 144

inverse in a Banach algebra, 35inverse operator, 39Inverse Operator Theorem, 56

Jacobson radical, 246James’ space, 86–100James’ Theorem, 134joint kernel, 120

K-theory, 188kernel, 18, 179Kreın–Milman Theorem, 140Kronecker symbol, 28Kuiper’s Theorem, 188

Lagrangian subspace, 314, 345Laplace operator, 298, 320linear functional

bounded, 17positive, 68

linear operatoradjoint, 165, 225bidual, 165bounded, 17closeable, 62closed, 59cokernel, 179compact, 174, 233completely continuous, 174complexified, 201

464 Index

cyclic vector, 281dissipative, 374dual, 164exponential map, 221finite rank, 174Fredholm, 179image, 179inverse, 39kernel, 179logarithm, 221normal, 227, 321positive semidefinite, 245projection, 78, 147right inverse, 78self-adjoint, 165, 227, 313singular value, 233spectrum, 208square root, 221, 245symmetric, 61, 63unitary, 227weakly compact, 190

linear subspaceclosure, 76complemented, 78dense, 76dual of, 76invariant, 288orthogonal complement, 225weak* closed, 130weak* dense, 122weakly closed, 119

Lipschitz continuous, 413long exact sequence, 195Lumer–Phillips Theorem, 375

Markov–KakutaniFixed Point Theorem, 161

maximal ideal, 246meagre, 40Mean Ergodic Theorem, 145measurable function

Banach space valued, 404strongly, 404weakly, 404

measurecomplex, 200ergodic, 144invariant, 125probability, 125projection valued, 262pushforward, 165signed, 4

spectral, 263metric space, 2

compact, 5complete, 3completion, 45

Milman–Pettis Theorem, 156Minkowski functional, 104

nonmeagre, 40norm, 3

equivalent, 18operator, 17

normal operator, 227spectrum, 229unbounded, 321

normed vector space, 3dual space, 26

weak* topology, 114product, 24quotient, 23strictly convex, 106uniformly convex, 156weak topology, 114

nowhere dense, 40

openball, 2half-space, 72map, 54set in a metric space, 2

Open Mapping Theorem, 54for unbounded operators, 343

operator norm, 17ordered vector space, 68orthogonal complement, 34

complex, 225orthonormal basis, 79

partial order, 445Pettis’ Lemma, 48Pettis’ Theorem, 405Phillips’ Lemma, 101Pitt’s Theorem, 191pointwise

bounded, 50compact, 12precompact, 12

polar set, 156positive cone, 68positive linear functional, 68pre-annihilator, 120precompact

Index 465

pointwise, 12subset of a metric space, 5subset of a topological space, 5

probability measure, 125product space, 25product topology, 110, 113, 450projection, 78, 147

quasi-seminorm, 65quotient space, 23

dual of, 76

Radon measure, 155Radon–Nikodym property, 415reflexive Banach space, 80–84, 134residual, 40resolvent

identity, 210, 300for semigroups, 368

operator, 210, 299set, 208, 299

Riemann–Lebesgue Lemma, 103Riesz Lemma, 22Ruston’s Theorem, 106

Schatten’s tensor product, 105Schauder basis, 99, 106, 178Schrodinger equation, 386Schrodinger operator, 298Schur’s Theorem, 121, 153second category, 40self-adjoint operator, 165, 227

spectrum, 231unbounded, 313

semigroupstrongly continuous, 350

seminorm, 65separable

Banach space, 85Hilbert space, 79topological space, 11

signed measure, 4total variation, 333

simplexinfinite-dimensional, 143

singular value, 233Smulyan–James Theorem, 159Snake Lemma, 195Sobolev space, 423spectral

measure, 263, 332projection, 215, 305

radius, 37, 39, 211Spectral Mapping Theorem

bounded linear operators, 217normal operators, 267self-adjoint operators, 240, 326unbounded operators, 326

Spectral Theorem, 281spectrum, 208

continuous, 208, 299in a unital Banach algebra, 246of a commutative algebra, 246of a compact operator, 213of a normal operator, 229, 324of a self-adjoint operator, 231of a unitary operator, 229of an unbounded operator, 299point, 208, 299residual, 208, 299

square root, 245, 344Babylonian method, 236

Stone’s Theorem, 384Stone–Weierstraß Theorem, 236, 289,

290strictly convex, 106, 143, 160strong convergence, 52strongly continuous semigroup, 350

analytic, 388–403contraction, 374dual semigroup, 377extension to a group, 366heat kernel, 352, 403Hille–Yosida–Phillips, 368infinitesimal generator, 357on a Hilbert space, 351regularity problem, 429Schrodinger equation, 386self-adjoint, 382shift operators, 351, 385unitary group, 384well-posed Cauchy problem, 363

symmetric linear operator, 61, 165symplectic

form, 314, 345reduction, 345vector space, 314, 345

tensor product, 105topological vector space, 110

locally convex, 110topology, 2

compact-open, 154of a metric space, 2

466 Index

of a normed vector space, 3product, 110, 113, 450strong, 110, 114strong operator, 52uniform operator, 17weak, 114weak*, 114

total variationof a signed measure, 333

totally bounded, 5triangle inequality, 2, 31trigonometric polynomial, 290Tychonoff’s Theorem, 450

unbounded operator, 295densely defined, 295normal, 321self-adjoint, 313spectral projection, 305spectrum, 299, 324with compact resolvent, 302

Uniform Boundedness Theorem, 50unitary operator, 227

spectrum, 229

vector spacecomplex normed, 198complexification, 201normed, 3ordered, 68topological, 110

Volterra operator, 291von Neumann’s Mean Ergodic

Theorem, 146

wave equation, 353, 387weak

compactness, 134–139continuity, 404convergence, 114measurability, 404topology, 114, 119–121

weak*compactness, 126, 127convergence, 114sequential closedness, 127sequential compactness, 124, 127topology, 114, 122–123, 130–133

weakly compact, 190winding number, 216

Zorn’s Lemma, 446

For additional information and updates on this book, visit

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GSM/191

Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction

It begins in Chapter 1 with an introduction to the necessary foundations, including the Arzelà–Ascoli theorem, elementary Hilbert space theory, and the Baire Category Theorem. Chapter 2 develops the three fundamental principles of functional anal-ysis (uniform boundedness, open mapping theorem, Hahn–Banach theorem) and

weak∗ topologies and includes the theorems of Banach–Alaoglu, Banach–Dieudonné, Eberlein–Smulyan, Krein–Milman, as well as an introduction to topological vector spaces and applications to ergodic theory. Chapter 4 is devoted to Fredholm theory. It includes an introduction to the dual operator and to compact operators, and it establishes the closed image theorem. Chapter 5 deals with the spectral theory of

continuous functional calculus for self-adjoint and normal operators, the Gelfand spectrum, spectral measures, cyclic vectors, and the spectral theorem. Chapter 6 introduces unbounded operators and their duals. It establishes the closed image

unbounded self-adjoint operators on Hilbert spaces. Chapter 7 gives an introduction -

measurable functions with values in a Banach space, and a discussion of solutions to

proof of Tychonoff’s theorem.

-

analysis in one variable, and measure and integration.

www.ams.org