Joan Cerdà

Linear Functional Analysis

Graduate Studies in Mathematics

Volume 116

American Mathematical SocietyReal Sociedad Matemática Española

Linear Functional Analysis

http://dx.doi.org/10.1090/gsm/116

Linear Functional Analysis

Joan Cerdà

Graduate Studies in Mathematics

Volume 116

American Mathematical Society Providence, Rhode Island

Real Sociedad Matemática Española Madrid, Spain

Editorial Board of Graduate Studies in Mathematics

David Cox (Chair)

Rafe Mazzeo Martin Scharlemann Gigliola Staffilani

Editorial Committee of the Real Sociedad Matematica Espanola

Guillermo P. Curbera, Director

Luis Alıas Alberto ElduqueEmilio Carrizosa Rosa Marıa MiroBernardo Cascales Pablo PedregalJavier Duoandikoetxea Juan Soler

2010 Mathematics Subject Classification. Primary 46–01; Secondary 46Axx, 46Bxx,46Exx, 46Fxx, 46Jxx, 47B15.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-116

Library of Congress Cataloging-in-Publication Data

Cerda, Joan, 1942–Linear functional analysis / Joan Cerda.

p. cm. — (Graduate studies in mathematics ; v. 116)Includes bibliographical references and index.ISBN 978-0-8218-5115-9 (alk. paper)1. Functional analysis. I. Title.

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10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10

To Carla and Marc

Contents

Preface xi

Chapter 1. Introduction 1

§1.1. Topological spaces 1

§1.2. Measure and integration 8

§1.3. Exercises 21

Chapter 2. Normed spaces and operators 25

§2.1. Banach spaces 26

§2.2. Linear operators 39

§2.3. Hilbert spaces 45

§2.4. Convolutions and summability kernels 52

§2.5. The Riesz-Thorin interpolation theorem 59

§2.6. Applications to linear differential equations 63

§2.7. Exercises 69

Chapter 3. Frechet spaces and Banach theorems 75

§3.1. Frechet spaces 76

§3.2. Banach theorems 82

§3.3. Exercises 88

Chapter 4. Duality 93

§4.1. The dual of a Hilbert space 93

§4.2. Applications of the Riesz representation theorem 98

§4.3. The Hahn-Banach theorem 106

vii

viii Contents

§4.4. Spectral theory of compact operators 114

§4.5. Exercises 122

Chapter 5. Weak topologies 127

§5.1. Weak convergence 127

§5.2. Weak and weak* topologies 128

§5.3. An application to the Dirichlet problem in the disc 132

§5.4. Exercises 138

Chapter 6. Distributions 143

§6.1. Test functions 144

§6.2. The distributions 146

§6.3. Differentiation of distributions 150

§6.4. Convolution of distributions 154

§6.5. Distributional differential equations 161

§6.6. Exercises 175

Chapter 7. Fourier transform and Sobolev spaces 181

§7.1. The Fourier integral 182

§7.2. The Schwartz class S 186

§7.3. Tempered distributions 189

§7.4. Fourier transform and signal theory 195

§7.5. The Dirichlet problem in the half-space 200

§7.6. Sobolev spaces 206

§7.7. Applications 213

§7.8. Exercises 222

Chapter 8. Banach algebras 227

§8.1. Definition and examples 228

§8.2. Spectrum 229

§8.3. Commutative Banach algebras 234

§8.4. C∗-algebras 238

§8.5. Spectral theory of bounded normal operators 241

§8.6. Exercises 250

Chapter 9. Unbounded operators in a Hilbert space 257

§9.1. Definitions and basic properties 258

§9.2. Unbounded self-adjoint operators 262

Contents ix

§9.3. Spectral representation of unbounded self-adjoint operators 273

§9.4. Unbounded operators in quantum mechanics 277

§9.5. Appendix: Proof of the spectral theorem 287

§9.6. Exercises 295

Hints to exercises 299

Bibliography 321

Index 325

Preface

The aim of this book is to present the basic facts of linear functional anal-ysis related to applications to some fundamental aspects of mathematicalanalysis.

If mathematics is supposed to show common general facts and struc-tures of particular results, functional analysis does this while dealing withclassical problems, many of them related to ordinary and partial differentialequations, integral equations, harmonic analysis, function theory, and thecalculus of variations.

In functional analysis, individual functions satisfying specific equationsare replaced by classes of functions and transforms which are determined byeach particular problem. The objects of functional analysis are spaces andoperators acting between them which, after systematic studies intertwininglinear and topological or metric structures, appear to be behind classicalproblems in a kind of cleaning process.

In order to make the scope of functional analysis clearer, I have chosento sacrifice generality for the sake of an easier understanding of its methods,and to show how they clarify what is essential in analytical problems. Ihave tried to avoid the introduction of cold abstractions and unnecessaryterminology in further developments and, when choosing the different topics,I have included some applications that connect functional analysis with otherareas.

The text is based on a graduate course taught at the Universitat deBarcelona, with some additions, mainly to make it more self-contained. Thematerial in the first chapters could be adapted as an introductory courseon functional analysis, aiming to present the role of duality in analysis, and

xi

xii Preface

also the spectral theory of compact linear operators in the context of Hilbertand Banach spaces.

In this first part of the book, the mutual influence between functionalanalysis and other areas of analysis is shown when studying duality, withvon Neumann’s proof of the Radon-Nikodym theorem based on the Rieszrepresentation theorem for the dual of a Hilbert space, followed by the rep-resentations of the duals of the Lp spaces and of C(K), in this case by meansof complex Borel measures.

The reader will also see how to deal with initial and boundary valueproblems in ordinary linear differential equations via the use of integraloperators. Moreover examples are included that illustrate how functionalanalytic methods are useful in the study of Fourier series.

In the second part, distributions provide a natural framework extend-ing some fundamental operations in analysis. Convolution and the Fouriertransform are included as useful tools for dealing with partial differentialoperators, with basic notions such as fundamental solutions and Green’sfunctions.

Distributions are also appropriate for the introduction of Sobolev spaces,which are very useful for the study of the solutions of partial differentialequations. A clear example is provided by the resolution of the Dirichletproblem and the description of the eigenvalues of the Laplacian, in combi-nation with Hilbert space techniques.

The last two chapters are essentially devoted to the spectral theory ofbounded and unbounded self-adjoint operators, which is presented by us-ing the Gelfand transform for Banach algebras. This spectral theory isillustrated with an introduction to the basic axioms of quantum mechanics,which motivated many studies in the Hilbert space theory.

Some very short historical comments have been included, mainly bymeans of footnotes. For a good overview of the evolution of functionalanalysis, J. Dieudonne’s and A. F. Monna’s books, [10] and [31], are twogood references.

The limitation of space has forced us to leave out many other importanttopics that could, and probably should, have been included. Among themare the geometry of Banach spaces, a general theory of locally convex spacesand structure theory of Frechet spaces, functional calculus of nonnormaloperators, groups and semigroups of operators, invariant subspaces, indextheory, von Neumann algebras, and scattering theory. Fortunately, manyexcellent texts dealing with these subjects are available and a few referenceshave been selected for further study.

Preface xiii

A small number of references have been gathered at the end of eachchapter to focus the reader’s attention on some appropriate items from ageneral bibliographical list of 44 items.

Almost 240 exercises are gathered at the end of the chapters and forman important part of the book. They are intended to help the reader todevelop techniques and working knowledge of functional analysis. Theseexercises are highly nonuniform in difficulty. Some are very simple, to aidin better understanding of the concepts employed,challenging for the beginners. Hints and solutions are provided at the endof the book.

The prerequisites are very standard. Although it is assumed that thereader has some a priori knowledge of general topology, integral calculus withLebesgue measure, and elementary aspects of normed or Hilbert spaces, areview of the basic aspects of these topics has been included in the firstchapters.

I turn finally to the pleasant task of thanking those who helped meduring the writing. Particular thanks are due to Javier Soria, who revisedmost of the manuscript and proposed important corrections and suggestions.I have also received valuable advice and criticism from Marıa J. Carro andJoaquim Ortega-Cerda. I have been very fortunate to have received theirassistance.

Joan CerdaUniversitat de Barcelona

whereas others are fairly

Bibliography

[1] R. A. Adams, Sobolev Spaces, Academic Press, Inc., 1975.

[2] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators inHilbert Space, Dover, 1993.

[3] S. Banach, Theorie des operations lineaires, Monografje Matamaty-czne, Warsaw, 1932, and Chelsea Publishing Co., 1955.

[4] S. K. Berberian, Lectures in Functional Analysis and Operator Theory,Springer-Verlag, 1974.

[5] H. Brezis, Analyse fonctionelle: Theorie et applications, Masson, 1983.

[6] J. Cerda, Analisis Real, Edicions de la Universitat de Barcelona, 1996.

[7] B. A. Conway, A Course in Functional Analysis, Springer-Verlag,1985.

[8] R. Courant and D. Hilbert, Methods of Mathematical Physics, OxfordUniversity Press, 1953.

[9] J. Dieudonne, Foundations of Modern Analysis, Academic Press, 1960.

[10] J. Dieudonne, History of Functional Analysis, North-Holland, 1981.

[11] P. A. M. Dirac, The Principles of Quantum Mechanics, ClarendonPress, 1930.

[12] N. Dunford and J. T. Schwartz, Linear Operators: Part 1, IntersciencePublishers, Inc., 1958.

[13] R. E. Edwards, Functional Analysis, Theory and Applications, Hold,Rinehart and Windston, 1965.

[14] G. B. Folland, Introduction to Partial Differential Equations, PrincetonUniversity Press and University of Tokyo Press, 1976.

321

322 Bibliography

[15] I. M. Gelfand, and G. E. Chilov, Generalized Functions, AcademicPress, New York, 1964 (translated from the 1960 Russian edition).

[16] I. M. Gelfand, D. A. Raikov and G. E. Chilov, Commutative NormedRings, Chelsea Publishing Co., New York, 1964 .

[17] D. Gilbarg and N. S.of Second Order, Springer-Verlag, 2001 (revision of the 1983 edition).

[18] P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., 1950.

[19] E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Amer.Math. Soc. Colloquium Publ., vol. 31, 1957.

[20] L. Hormander, Linear Partial Differential Operators, Springer-Verlag,1963.

[21] J. Horvath, Topological Vector Spaces and Distributions, Addison Wes-ley, 1966.

[22] L. Kantorovitch and G. Akilov, Analyse fonctionnelle, “Mir”, 1981(translated from the 1977 Russian edition).

[23] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag,1976.

[24] J. L. Kelley, General Topology, Van Nostrand, Princeton, NJ, 1963.

[25] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Func-tions and Functional Analysis, Glaglock Press 1961 (translated fromthe 1960 Russian edition).

[26] G. Kothe, Topological Vector Spaces I, Springer-Verlag, 1969.

[27] P. D. Lax, Functional Analysis, Wiley, 2002.

[28] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics,American Mathematical Society, 1997.

[29] V. Mazya, Sobolev Spaces, Springer-Verlag, 1985.

[30] R. Meise and D. Vogdt, Introduction to Functional Analysis, TheClarendon Press, Oxford University Press, 1997 (translated from the1992 German edition).

[31] A. F. Monna, Functional Analysis in Historical Perspective, John Wiley& Sons, 1973.

[32] M. A. Naimark, Normed Rings, Erven P. Noordhoff, Ltd., 1960 (trans-lation of the 1955 Rusian edition).

[33] E. Prugovecki, Quantum Mechanics in Hilbert Space, Academic Press,1971.

[34] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Aca-demic Press, 1972.

[35] C. E. Rickart, General Theory of Banach Algebras, D. Van NostrandCompany, Inc., 1960.

Trudinger, Elliptic Partial Differential Equations

Bibliography 323

[36] F. Riesz and B. Sz. Nagy, Lecons d’analyse fonctionelle, AkademiaiKiado, Budapest, 1952, and Functional Analysis, F. Ungar, 1955.

[37] H. L. Royden, Real Analysis, The Macmillan Company, 1968.

[38] W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.

[39] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Company,1966.

[40] L. Schwartz, Theorie des distributions, Hermann & Cie, 1969 (newedition, 1973).

[41] E. M. Stein and G.Spaces, Princeton University Press, 1971.

[42] A. E. Taylor and D.Wiley & Sons, Inc., second edition, 1980.

[43] J. von Neumann, Mathematical Foundations of Quantum Mechanics,Princeton University Press, 1996 (translation of the 1932 German edi-tion).

[44] K. Yosida, Functional Analysis, Springer-Verlag, 1968.

Weiss, Introduction to Fourier Analysis on Euclidean

C. Lay, Introduction to Functional Analysis, John

Index

⊎, 9

≺, 8

∂j , 30

Δ(A), 234

δa, δ, 148

χA, 10

σ-additivity, 9

σ-algebra, 8

σ(E, E), 129σ(T ), 118

σA(a), 229

τh, 33, 150

A(D), 229, 251

A⊥, 47

Ao, 112

Aψ, 280

BE , 115

B(X), 29, 229

BΩ, 9

C∗-algebra, 238C(K), 33, 229

Cm(Ω), 30

C0(Rn), 70, 189

Cb(X), 30

Cc(X), 7

co, 123

c0, 59

c00, 69

Dα, 30, 151

D′(Ω), 147

D(Ω), 145

Dm(Ω), 212

DK(Ω), 81

DmK (Ω), 82

E-almost everywhere (E-a.e.), 246

E-essential supremum (E- sup), 246

eξ, 182

E′(Ω), 156

E(Ω), 81

Em(Ω), 82

H∞, 236

Hm(Ω), 210

Hm0 (Ω), 212

Hs, 208

H(Ω), 88

L1loc, 145

L∞, 229

L∞(E), 287

Lp, 14, 31, 100

Lp(T), 56

LpT (R), 56

L(E), 43

L(E;F ), 42

Lc(E), 115

Lc(E;F ), 115

�, 198

�∞, 29

�p, 31, 103

Ra(λ), 230

r(a), 232

S(X), 60

sinc , 183

suppE, 246

supp f , 7

suppu, 155

S, 186S′(Rn), 189

varψ(A), 280

W , 253

Wm,p(Ω), 206

325

326 Index

Alaoglu theorem, 130Alaoglu, L., 130Algebra

Banach, 228Banach unitary, 228

disc, 229, 251uniform, 253Wiener, 253

Alias, 198Almost everywhere (a.e.), 11

Annihilation operator, 285Annihilator of a subset, 112Approximation of the identity, 54, 56Arzela, C., 115Ascoli, G., 115Ascoli-Arzela theorem, 115

Baire’s theorem, 82Ball, 1Banach limit, 122Banach, S., 28, 83, 85, 103, 106, 130

Banach-Schauder theorem, 83Banach-Steinhaus theorem, 86Band, 195Bernstein polynomials, 35Bernstein, S. N., 35

Bessel’s inequality, 49Bilinear form, 45Bohnenblust, H. F., 107Borel σ-algebra, 9

Borel, E., 10Born, M., 257, 282Bound states, 287Boundary

conditions, separated, 66problem, 68

value problem, 66Bourbaki, N., 14Bunyakovsky, V., 45Buskes, G., 107

Calderon, A., 59Canonical

commutation relations, 297isomorphism, 234

Canonically

conjugate, 297Caratheodory condition, 16Caratheodory, C., 16Carleson, L., 193, 238Cauchy

integral, 125

problem, 64, 71sequence, 4

Cauchy, A., 45Cauchy-Bunyakovsky-Schwarz inequality,

45

Cayley transform, 293Cayley, A., 114, 293Cech, E., 6Cesaro sums, 58

Chain rule, 211Character, 234Closed graph theorem, 84Closure of a set, 2Coercive sesquilinear form, 95

Commutator, 261, 282, 283, 297Commuting

operators, 283spectral measures, 283

Compact linear map, 115

Completion, 113of a normed space, 33

Complex spectral measures, 242Convex

functional, 106

hull, 88Convolution, 53, 56, 154Convolvable functions, 53Corona problem, 237Creation operator, 285

Cyclic vector, 296

Daniel, P. J., 14de la Vallee-Poussin kernel, 72Dieudonne, J., 77Dilation, 150, 191Dirac comb, 176

Dirac, P., 143, 257Dirichlet

kernel, 57norm, 215

Dirichlet problem, 135, 200, 214, 216, 220

in the ball, 172weak solution, 215

Dirichlet, J., 200Distance, 1Distribution, 146, 147

Dirac, 148tempered, 190with compact support, 156

Distribution function, 21Distribution of an observable, 280

Distributionalconvergence, 148derivative, 151

Divergence theorem, 163Domain of an operator, 258

Dominated convergence theorem, 12Dual couple, 128

Eigenspace, 118Eigenvalue, 118, 259

approximate, 122, 264

Index 327

Eigenvector, 118Energy eigenvalues, 287Equicontinuous, 115Essential range, 252

Euclideandistance, 1inner product, 47norm, 2

Evolution operator, 281

Expected value, 280

Fatoulemma, 11theorem, 136

Fatou, P., 11Fejer kernel, 57

Finite intersection property, 4Fischer, E., 50Fischer-Riesz theorem, 50Fourier

coefficients, 50, 56

integral, 182multiplier, 208series, 50, 51, 56transform, 182transform, inverse, 183

Frechet norm, 79Frechet, M., 3, 28, 80Fredholm

alternative, 118operator, 39, 117

Fredholm, E.I., 39Friedrichs

extension, 271theorem, 211

Friedrichs, K., 272

Functionabsolutely continuous, 12absolutely integrable, 12continuous, 3Dirac, 143

Heaviside, Y , 152integrable, 12locally integrable, 145, 147measurable, 11periodized, 186

sequentially continuous, 4test, 145

Functional calculus, 241Fundamental solution, 162

Gauge, 108Gauss-Weierstrass kernel, 73, 184, 222

Gelfandtopology, 235transform, 236

Gelfand, I., 227, 232, 234

Gelfand-Mazur theorem, 234Gelfand-Naimark theorem, 240Goldstine theorem, 139Graph, 260

Green’sfunction, 67, 170, 171identities, 163

Green, G., 67, 163

Ground state, 287

Holder inequality, 13

Holder, O., 13Hahn, H., 85, 106Hahn-Banach theorem, 106–108, 110Hamiltonian, 279, 280

Hardy, G.H., 204Harmonic oscillator

classical, 284quantic, 285

Hausdorff, F., 2, 3Heat equation, 166, 184, 222Heaviside, O., 143Heine-Borel

property, 79theorem, 5

Heisenberg, W., 257Helly, E., 106

Herglotz theorem, 136Hermite polynomial, 286Hermitian element, 239Hilbert transform, 204

Hilbert, D., 118, 121, 204, 247Hilbert-Schmidt

operator, 117spectral theorem, 120

Homomorphismof C∗-algebras, 239of unitary Banach algebras, 229

Ideal, 234maximal, 234

Incompatible observables, 283Infinitesimal generator, 281

Initialvalue problem, 64

Inner product, 45Inner regularity, 10

Integral kernel, 52Interior of a set, 2Internal point, 108Inversion theorem, 188

Involution, 238Isomorphism of normed spaces, 40

Jordan, P., 257

Kakutani, S., 103

328 Index

Kinetic energy, 278Kondrachov, V., 218

Lagrange identity, 64Laplacian, 163Lax, P., 95

Lax-Milgram theorem, 95Lebesgue

differentiation theorem, 12integral, 11

numbers, 87Lebesgue, H., 10Leibniz’s formula, 30Leray, J., 181Local

basis, 27subbasis, 77

Local uniform convergence, 77

Malgrange-Ehrenpreis theorem, 162Mazur, S., 234

Mean value, 280Measure, 9

σ-finite, 10absolutely continuous, 98

Borel, 10complex, 18counting, 22, 31, 47Dirac, 10Lebesgue, 10

outer, 15positive, 9Radon, 14real, 18

regular, 10spectral, 244

Metric, 1Meyer-Serrin theorem, 211

Minkowskifunctional, 88, 108inequality, 13integral inequality, 125

Minkowski, H., 13

Modulation, 186, 191Mollifier, 145Momentum, 277Monotone convergence theorem, 11

Multiplication by a C∞ function, 148Multiplicity of an eigenvalue, 118

Nagumo, M., 227Nearly orthogonal, 41Neighborhood, 2

basis, 2Nelson, E., 283Neumann series, 43, 231, 260Neumann, C., 43

Norm, 27equivalent, 40finer, 40

Normal element, 239

Nyquistfrequency, 195rate, 198

Nyquist, H., 195

Observable, 279values, 280

Open mapping theorem, 83Operator

adjoint, 96, 261bounded linear , 39Cauchy-Riemann, 178

closable, 260closed, 260compact, 115densely defined, 261derivative, 258

differential, 162elliptic, 162energy, 279essentially self-adjoint, 270heat, 165

Hilbert-Schmidt, 97hypoelliptic, 162momentum, 278norm, 42position, 259

relatively bounded, 267self-adjoint, 97, 263semi-bounded, 272symmetric, 263unitary, 248

wave, 168Order

of a distribution, 147of a differential operator, 30

Orthogonal complement, 49

Orthonormalbasis, 50system, 49

Outer regularity, 10

Parallelogram identity, 47Parseval’s relation, 50

Parseval, M. A., 52Partition of unity, 8Periodic extension, 186Plancherel, M., 192Planck constant, 278

Planck, M., 278, 287Poincare lemma, 215Poisson

equation, 165

Index 329

integral, 135, 172kernel, 55, 172, 173, 201kernel of the disc, 134kernel, conjugate, 203

theorem, 184Poisson, S. D., 165, 184Polar decomposition of a bounded normal

operator, 254Polar representation of a complex measure,

104

Polarization identities, 51, 98Position, 277Positive linear form, 12, 100Product norm, 30

Projectionoptimal, 47orthogonal, 49theorem, 47

Pythagorean theorem, 47

Quadrature method, 90Quaternions, 252Quotient

locally convex space, 78

map, 78space, 112

Radon, J., 14, 103Radon-Nikodym

derivative, 99

theorem, 98Rayleigh, Lord, 192Regular point, 259Regularization, 176Rellich, F., 218, 268

Resolution of the identity, 244Resolvent, 230

of an operator, 260Rickart, C., 227

Riemann-Lebesgue lemma, 59, 189Riesz

lemma, 41representation theorem, 94

for (Lp)′, 101for C(K)′, 104

theorem for the Hilbert transform, 205Riesz, F., 14, 50, 103, 118, 247

Riesz, M., 59, 206Riesz-Markov representation theorem, 15Riesz-Thorin theorem, 61, 73Rogers, L., 13

Sampling, 195

Scalar product, 45, 93Schauder theorem, 116Schauder, J., 83, 116Schmidt, E., 46, 121

Schrodingerequation, 282picture, 279

Schrodinger, E., 257, 282Schur, I., 59Schwartz, L., 14, 77, 144Schwarz inequality, 45Schwarz, H., 45Self-adjoint

element, 239form, 63

Semi-norm, 76Separated sets, 109

strictly, 109Sesquilinear form, 45Set

Gδ, 82absorbing, 76, 108

balanced, 76Borel, 9bounded, 39, 78closed, 2compact, 4convex, 27measurable, 9open, 2resolvent, 259

Shannonsystem, 196theorem, 196

Shannon, C., 196Signal

analog-to-digital conversion, 195band-limited, 195continuous time, 195discrete time, 195

Skewlinear, 45Slowly increasing sequence, 198Sobczyk, A., 107Sobolev, S., 181Space

Banach, 28compact, 4complete, 5countably semi-normable, 79dual, 43Euclidean, 1, 29, 47finite-dimensional, 40Frechet, 80Hausdorff, 2Hilbert, 46, 93locally compact, 7locally convex , 77measurable, 8measure, 9metric, 1normable, 28

330 Index

normed, 28reflexive normed, 139separable, 36sequentially compact, 4Sobolev, 206, 208

topological, 2Spectral radius, 232Spectrum, 118, 198, 229, 259

continuous, 259of a commutative unitary Banach

algebra, 234point, 259residual, 259

Square wave, 183, 199State, 279

space, 279Steinhaus, H., 85Stone’s theorem, 281Stone, M., 37, 272

Stone-Weierstrass theorem, 37complex form, 38

Strongly continuous one-parameter groupof unitary operators, 281

Sturm-Liouville problem, 213classical solution, 213weak solution, 213

Subspace, 27, 29complemented, 111orthogonal, 47topological, 3

Sufficient family of semi-norms, 76Summability kernel, 54, 56Support

of a distribution, 155of a function, 7of a spectral measure, 246

Sylvester, J., 114Symmetry, 150, 191

Taylor, A., 232Thorin, O., 59Three lines theorem, 59Topological vector space, 26Topologically complementary subspaces,

111

Topology, 2coarser, 3finer, 3metrizable, 4of a metric space, 3product, 6weak, 129

weak*, 130Total subset, 36Translation, 150, 191Transpose, 96, 113Transposition map, 114

Tychonoff, A. N. or Tichonov, A. N., 6Type (p, q), 52

Uncertainty principle, 195, 282Uniform boundedness principle, 85Uniform norm, 29Uniqueness theorem for Fourier coefficients,

51, 59Unitarily equivalent operators, 259Urysohn function

smooth, 145continuous, 7

Variance, 280Variation

positive, negative, 20total, 18

Vector topology, 26Volterra operator, 40, 117Volterra, V., 40von Neumann, J., 46, 257, 272, 293

Wave equation, 179Wave function, 279Weak derivative, 151Weierstrass theorem, 34, 35Weierstrass, K., 34

Weyl’s lemma, 165Whitney’s notation, 30Wintner, A., 272Wronskian, 65Wronskian determinant, 64

Young’s inequality, 54, 63Young, W. H., 10, 14

Zorn’s lemma, 6, 50, 107

GSM/116

Functional analysis studies the algebraic, geometric, and topo-logical structures of spaces and operators that underlie many classical problems. Individual functions satisfying specific equations are replaced by classes of functions and transforms that are deter-mined by the particular problems at hand.

This book presents the basic facts of linear functional analysis as related to fundamental aspects of mathematical analysis and their applications. The exposition avoids unnecessary terminology and generality and focuses on showing how the knowledge of these structures clarifies what is essential in analytic problems.

The material in the first part of the book can be used for an introductory course on functional analysis, with an emphasis on the role of duality. The second part introduces distributions and Sobolev spaces and their applications. Convolution and the Fourier transform are shown to be useful tools for the study of partial differential equations. Fundamental solutions and Green’s functions are considered and the theory is illustrated with several applications. In the last chapters, the Gelfand transform for Banach algebras is used to present the spectral theory of bounded and unbounded operators, which is then used in an introduction to the basic axioms of quantum mechanics.

The presentation is intended to be accessible to readers whose backgrounds include basic linear algebra, integration theory, and general topology. Almost 240 exercises will help the reader in better understanding the concepts employed.

American Mathematical Society www.ams.org

Real Sociedad Matemática Española www.rsme.es

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-116