Linear Functional Analysis

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Joan Cerdà

Graduate Studies in Mathematics Volume 116

American Mathematical Society Real Sociedad Matemática Española 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 Matemática Española Guillermo P. Curbera, Director Luis Al´ıas Alberto Elduque Emilio Carrizosa Rosa Mar´ıa Miró Bernardo Cascales Pablo Pedregal Javier 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, visit www.ams.org/bookpages/gsm-116

Library of Congress Cataloging-in-Publication Data Cerd`a, Joan, 1942– Linear functional analysis / Joan Cerd`a. 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. QA321.C47 2010 515.7—dc22 2010006449

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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. Fr´echet spaces and Banach theorems 75 §3.1. Fr´echet 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 analysis related to applications to some fundamental aspects of mathematical analysis. If mathematics is supposed to show common general facts and structures of particular results, functional analysis does this while dealing with classical problems, many of them related to ordinary and partial differential equations, integral equations, harmonic analysis, function theory, and the calculus of variations. In functional analysis, individual functions satisfying specific equations are replaced by classes of functions and transforms which are determined by each particular problem. The objects of functional analysis are spaces and operators acting between them which, after systematic studies intertwining linear and topological or metric structures, appear to be behind classical problems in a kind of cleaning process. In order to make the scope of functional analysis clearer, I have chosen to sacrifice generality for the sake of an easier understanding of its methods, and to show how they clarify what is essential in analytical problems. I have tried to avoid the introduction of cold abstractions and unnecessary terminology in further developments and, when choosing the different topics, I have included some applications that connect functional analysis with other areas. The text is based on a graduate course taught at the Universitat de Barcelona, with some additions, mainly to make it more self-contained. The material in the first chapters could be adapted as an introductory course on 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 Hilbert and Banach spaces. In this first part of the book, the mutual influence between functional analysis and other areas of analysis is shown when studying duality, with von Neumann’s proof of the Radon-Nikodym theorem based on the Riesz representation 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 means of complex Borel measures. The reader will also see how to deal with initial and boundary value problems in ordinary linear differential equations via the use of integral operators. Moreover examples are included that illustrate how functional analytic 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 Fourier transform are included as useful tools for dealing with partial differential operators, with basic notions such as fundamental solutions and Green’s functions. Distributions are also appropriate for the introduction of Sobolev spaces, which are very useful for the study of the solutions of partial differential equations. A clear example is provided by the resolution of the Dirichlet problem 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 of bounded and unbounded self-adjoint operators, which is presented by us- ing the Gelfand transform for Banach algebras. This spectral theory is illustrated 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 by means of footnotes. For a good overview of the evolution of functional analysis, J. Dieudonné’s and A. F. Monna’s books, [10] and [31], are two good references. The limitation of space has forced us to leave out many other important topics that could, and probably should, have been included. Among them are the geometry of Banach spaces, a general theory of locally convex spaces and structure theory of Fréchet spaces, functional calculus of nonnormal operators, groups and semigroups of operators, invariant subspaces, index theory, von Neumann algebras, and scattering theory. Fortunately, many excellent texts dealing with these subjects are available and a few references have been selected for further study. Preface xiii

A small number of references have been gathered at the end of each chapter to focus the reader’s attention on some appropriate items from a general bibliographical list of 44 items. Almost 240 exercises are gathered at the end of the chapters and form an important part of the book. They are intended to help the reader to develop techniques and working knowledge of functional analysis. These exercises are highly nonuniform in difficulty. Some are very simple, to aid in better understanding of the concepts employed, whereas others are fairly challenging for the beginners. Hints and solutions are provided at the end of the book. The prerequisites are very standard. Although it is assumed that the reader has some a priori knowledge of general topology, integral calculus with Lebesgue measure, and elementary aspects of normed or Hilbert spaces, a review of the basic aspects of these topics has been included in the first chapters. I turn finally to the pleasant task of thanking those who helped me during the writing. Particular thanks are due to Javier Soria, who revised most of the manuscript and proposed important corrections and suggestions. I have also received valuable advice and criticism from Mar´ıa J. Carro and Joaquim Ortega-Cerdà. I have been very fortunate to have received their assistance.

Joan Cerd`a Universitat de Barcelona

Bibliography

[1] R. A. Adams, Sobolev Spaces, Academic Press, Inc., 1975. [2] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, 1993. [3] S. Banach, Théorie des opérations linéaires, 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: Théorie et applications, Masson, 1983. [6] J. Cerdà, Análisis 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, Oxford University Press, 1953. [9] J. Dieudonné, Foundations of Modern Analysis, Academic Press, 1960. [10] J. Dieudonné, History of Functional Analysis, North-Holland, 1981. [11] P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, 1930. [12] N. Dunford and J. T. Schwartz, Linear Operators: Part 1, Interscience Publishers, 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, Princeton University Press and University of Tokyo Press, 1976.

321 322 Bibliography

[15] I. M. Gelfand, and G. E. Chilov, Generalized Functions, Academic Press, New York, 1964 (translated from the 1960 Russian edition). [16] I. M. Gelfand, D. A. Raikov and G. E. Chilov, Commutative Normed Rings, Chelsea Publishing Co., New York, 1964 . [17] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations 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. Hörmander, 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 from the 1960 Russian edition). [26] G. Köthe, 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, The Clarendon Press, Oxford University Press, 1997 (translated from the 1992 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 (translation of the 1955 Rusian edition). [33] E. Prugoveˇcki, 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 Nostrand Company, Inc., 1960. Bibliography 323

[36] F. Riesz and B. Sz. Nagy, Le¸cons d’analyse fonctionelle, Akadémiai Kiadó, 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, Théorie des distributions, Hermann & Cie, 1969 (new edition, 1973). [41] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. [42] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, Inc., second edition, 1980. [43] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1996 (translation of the 1932 German edition). [44] K. Yosida, Functional Analysis, Springer-Verlag, 1968.

Index

,9 E-almost everywhere (E-a.e.), 246 ≺,8 E-essential supremum (E- sup), 246 ∂j ,30 eξ, 182 E(Ω), 156 E Δ(A), 234 (Ω), 81 Em(Ω), 82 δa,δ, 148 H∞, 236 χA,10 m σ-additivity, 9 H (Ω), 210 m σ-algebra, 8 H0 (Ω), 212 σ(E,E), 129 Hs, 208 σ(T ), 118 H(Ω), 88 1 σA(a), 229 Lloc, 145 ∞ τh, 33, 150 L , 229 L∞(E), 287 p A(D), 229, 251 L , 14, 31, 100 ⊥ Lp(T), 56 A ,47 p Ao, 112 LT (R), 56 L A , 280 (E), 43 ψ L B , 115 (E; F ), 42 E L B(X), 29, 229 c(E), 115 Lc(E; F ), 115 BΩ,9 C∗-algebra, 238 , 198 ∞ C(K), 33, 229 ,29 p Cm(Ω),¯ 30 , 31, 103 n C0(R ), 70, 189 Ra(λ), 230 Cb(X), 30 r(a), 232 Cc(X), 7 S(X), 60 co, 123 sinc , 183 c0,59 supp E, 246 c00,69 supp f,7 Dα, 30, 151 supp u, 155 D(Ω), 147 S, 186 D(Ω), 145 S(Rn), 189 m D (Ω), 212 varψ(A), 280 DK (Ω), 81 W , 253 Dm m,p K (Ω), 82 W (Ω), 206

325 326 Index

Alaoglu theorem, 130 Cayley transform, 293 Alaoglu, L., 130 Cayley, A., 114, 293 Algebra Cech,ˇ E., 6 Banach, 228 Cesàro sums, 58 Banach unitary, 228 Chain rule, 211 disc, 229, 251 Character, 234 uniform, 253 Closed graph theorem, 84 Wiener, 253 Closure of a set, 2 Alias, 198 Coercive sesquilinear form, 95 Almost everywhere (a.e.), 11 Commutator, 261, 282, 283, 297 Annihilation operator, 285 Commuting Annihilator of a subset, 112 operators, 283 Approximation of the identity, 54, 56 spectral measures, 283 Arzelà, C., 115 Compact linear map, 115 Ascoli, G., 115 Completion, 113 Ascoli-Arzelà theorem, 115 of a normed space, 33 Complex spectral measures, 242 Baire’s theorem, 82 Convex Ball, 1 functional, 106 Banach limit, 122 hull, 88 Banach, S., 28, 83, 85, 103, 106, 130 Convolution, 53, 56, 154 Banach-Schauder theorem, 83 Convolvable functions, 53 Banach-Steinhaus theorem, 86 Corona problem, 237 Band, 195 Creation operator, 285 Bernstein polynomials, 35 Cyclic vector, 296 Bernstein, S. N., 35 Bessel’s inequality, 49 Daniel, P. J., 14 Bilinear form, 45 de la Vallée-Poussin kernel, 72 Bohnenblust, H. F., 107 Dieudonné, J., 77 Borel σ-algebra, 9 Dilation, 150, 191 Borel, E.,´ 10 Dirac comb, 176 Born, M., 257, 282 Dirac, P., 143, 257 Bound states, 287 Dirichlet Boundary kernel, 57 conditions, separated, 66 norm, 215 problem, 68 Dirichlet problem, 135, 200, 214, 216, 220 value problem, 66 in the ball, 172 Bourbaki, N., 14 weak solution, 215 Bunyakovsky, V., 45 Dirichlet, J., 200 Buskes, G., 107 Distance, 1 Distribution, 146, 147 Calderón, A., 59 Dirac, 148 Canonical tempered, 190 commutation relations, 297 with compact support, 156 isomorphism, 234 Distribution function, 21 Canonically Distribution of an observable, 280 conjugate, 297 Distributional Carathéodory condition, 16 convergence, 148 Carathéodory, C., 16 derivative, 151 Carleson, L., 193, 238 Divergence theorem, 163 Cauchy Domain of an operator, 258 integral, 125 Dominated convergence theorem, 12 problem, 64, 71 Dual couple, 128 sequence, 4 Cauchy, A., 45 Eigenspace, 118 Cauchy-Bunyakovsky-Schwarz inequality, Eigenvalue, 118, 259 45 approximate, 122, 264 Index 327

Eigenvector, 118 Gelfand-Mazur theorem, 234 Energy eigenvalues, 287 Gelfand-Naimark theorem, 240 Equicontinuous, 115 Goldstine theorem, 139 Essential range, 252 Graph, 260 Euclidean Green’s distance, 1 function, 67, 170, 171 inner product, 47 identities, 163 norm, 2 Green, G., 67, 163 Evolution operator, 281 Ground state, 287 Expected value, 280 Hölder inequality, 13 Fatou Hölder, O., 13 lemma, 11 Hahn, H., 85, 106 theorem, 136 Hahn-Banach theorem, 106–108, 110 Fatou, P., 11 Hamiltonian, 279, 280 Fejér kernel, 57 Hardy, G.H., 204 Finite intersection property, 4 Harmonic oscillator Fischer, E., 50 classical, 284 Fischer-Riesz theorem, 50 quantic, 285 Fourier Hausdorff, F., 2, 3 coefficients, 50, 56 Heat equation, 166, 184, 222 integral, 182 Heaviside, O., 143 multiplier, 208 Heine-Borel series, 50, 51, 56 property, 79 transform, 182 theorem, 5 transform, inverse, 183 Heisenberg, W., 257 Fréchet norm, 79 Helly, E., 106 Fréchet, M., 3, 28, 80 Herglotz theorem, 136 Fredholm Hermite polynomial, 286 alternative, 118 Hermitian element, 239 operator, 39, 117 Hilbert transform, 204 Fredholm, E.I., 39 Hilbert, D., 118, 121, 204, 247 Friedrichs Hilbert-Schmidt extension, 271 operator, 117 theorem, 211 spectral theorem, 120 Friedrichs, K., 272 Homomorphism Function of C∗-algebras, 239 absolutely continuous, 12 of unitary Banach algebras, 229 absolutely integrable, 12 continuous, 3 Ideal, 234 Dirac, 143 maximal, 234 Heaviside, Y , 152 Incompatible observables, 283 integrable, 12 Infinitesimal generator, 281 locally integrable, 145, 147 Initial measurable, 11 value problem, 64 periodized, 186 Inner product, 45 sequentially continuous, 4 Inner regularity, 10 test, 145 Integral kernel, 52 Functional calculus, 241 Interior of a set, 2 Fundamental solution, 162 Internal point, 108 Inversion theorem, 188 Gauge, 108 Involution, 238 Gauss-Weierstrass kernel, 73, 184, 222 Isomorphism of normed spaces, 40 Gelfand topology, 235 Jordan, P., 257 transform, 236 Gelfand, I., 227, 232, 234 Kakutani, S., 103 328 Index

Kinetic energy, 278 Norm, 27 Kondrachov, V., 218 equivalent, 40 finer, 40 Lagrange identity, 64 Normal element, 239 Laplacian, 163 Nyquist Lax, P., 95 frequency, 195 Lax-Milgram theorem, 95 rate, 198 Lebesgue Nyquist, H., 195 differentiation theorem, 12 integral, 11 Observable, 279 numbers, 87 values, 280 Lebesgue, H., 10 Open mapping theorem, 83 Leibniz’s formula, 30 Operator Leray, J., 181 adjoint, 96, 261 Local bounded linear , 39 basis, 27 Cauchy-Riemann, 178 subbasis, 77 closable, 260 Local uniform convergence, 77 closed, 260 compact, 115 Malgrange-Ehrenpreis theorem, 162 densely defined, 261 Mazur, S., 234 derivative, 258 Mean value, 280 differential, 162 Measure, 9 elliptic, 162 σ-finite, 10 energy, 279 absolutely continuous, 98 essentially self-adjoint, 270 Borel, 10 heat, 165 complex, 18 Hilbert-Schmidt, 97 counting, 22, 31, 47 hypoelliptic, 162 Dirac, 10 momentum, 278 Lebesgue, 10 norm, 42 outer, 15 position, 259 positive, 9 relatively bounded, 267 Radon, 14 self-adjoint, 97, 263 real, 18 semi-bounded, 272 regular, 10 symmetric, 263 spectral, 244 unitary, 248 Metric, 1 wave, 168 Meyer-Serrin theorem, 211 Order Minkowski of a distribution, 147 functional, 88, 108 of a differential operator, 30 inequality, 13 Orthogonal complement, 49 integral inequality, 125 Orthonormal Minkowski, H., 13 basis, 50 Modulation, 186, 191 system, 49 Mollifier, 145 Outer regularity, 10 Momentum, 277 Monotone convergence theorem, 11 Parallelogram identity, 47 Multiplication by a C∞ function, 148 Parseval’s relation, 50 Multiplicity of an eigenvalue, 118 Parseval,M.A.,52 Partition of unity, 8 Nagumo, M., 227 Periodic extension, 186 Nearly orthogonal, 41 Plancherel, M., 192 Neighborhood, 2 Planck constant, 278 basis, 2 Planck, M., 278, 287 Nelson, E., 283 Poincaré lemma, 215 Neumann series, 43, 231, 260 Poisson Neumann, C., 43 equation, 165 Index 329

integral, 135, 172 Schrödinger kernel, 55, 172, 173, 201 equation, 282 kernel of the disc, 134 picture, 279 kernel, conjugate, 203 Schrödinger, E., 257, 282 theorem, 184 Schur, I., 59 Poisson, S. D., 165, 184 Schwartz, L., 14, 77, 144 Polar decomposition of a bounded normal Schwarz inequality, 45 operator, 254 Schwarz, H., 45 Polar representation of a complex measure, Self-adjoint 104 element, 239 Polarization identities, 51, 98 form, 63 Position, 277 Semi-norm, 76 Positive linear form, 12, 100 Separated sets, 109 Product norm, 30 strictly, 109 Projection Sesquilinear form, 45 optimal, 47 Set orthogonal, 49 Gδ,82 theorem, 47 absorbing, 76, 108 Pythagorean theorem, 47 balanced, 76 Borel, 9 Quadrature method, 90 bounded, 39, 78 Quaternions, 252 closed, 2 Quotient compact, 4 locally convex space, 78 convex, 27 map, 78 measurable, 9 space, 112 open, 2 resolvent, 259 Radon, J., 14, 103 Shannon Radon-Nikodym system, 196 derivative, 99 theorem, 196 theorem, 98 Shannon, C., 196 Rayleigh, Lord, 192 Signal Regular point, 259 analog-to-digital conversion, 195 Regularization, 176 band-limited, 195 Rellich, F., 218, 268 continuous time, 195 Resolution of the identity, 244 discrete time, 195 Resolvent, 230 Skewlinear, 45 of an operator, 260 Slowly increasing sequence, 198 Rickart, C., 227 Sobczyk, A., 107 Riemann-Lebesgue lemma, 59, 189 Sobolev, S., 181 Riesz Space lemma, 41 Banach, 28 representation theorem, 94 compact, 4 for (Lp), 101 complete, 5 for C(K), 104 countably semi-normable, 79 theorem for the Hilbert transform, 205 dual, 43 Riesz, F., 14, 50, 103, 118, 247 Euclidean, 1, 29, 47 Riesz, M., 59, 206 finite-dimensional, 40 Riesz-Markov representation theorem, 15 Fréchet, 80 Riesz-Thorin theorem, 61, 73 Hausdorff, 2 Rogers, L., 13 Hilbert, 46, 93 locally compact, 7 Sampling, 195 locally convex , 77 Scalar product, 45, 93 measurable, 8 Schauder theorem, 116 measure, 9 Schauder, J., 83, 116 metric, 1 Schmidt, E., 46, 121 normable, 28 330 Index

normed, 28 Tychonoff,A.N.orTichonov,A.N.,6 reflexive normed, 139 Type (p, q), 52 separable, 36 sequentially compact, 4 Uncertainty principle, 195, 282 Sobolev, 206, 208 Uniform boundedness principle, 85 topological, 2 Uniform norm, 29 Spectral radius, 232 Uniqueness theorem for Fourier coefficients, Spectrum, 118, 198, 229, 259 51, 59 continuous, 259 Unitarily equivalent operators, 259 of a commutative unitary Banach Urysohn function algebra, 234 smooth, 145 point, 259 continuous, 7 residual, 259 Square wave, 183, 199 Variance, 280 State, 279 Variation space, 279 positive, negative, 20 Steinhaus, H., 85 total, 18 Stone’s theorem, 281 Vector topology, 26 Stone, M., 37, 272 Volterra operator, 40, 117 Stone-Weierstrass theorem, 37 Volterra, V., 40 complex form, 38 von Neumann, J., 46, 257, 272, 293 Strongly continuous one-parameter group of unitary operators, 281 Wave equation, 179 Sturm-Liouville problem, 213 Wave function, 279 classical solution, 213 Weak derivative, 151 weak solution, 213 Weierstrass theorem, 34, 35 Subspace, 27, 29 Weierstrass, K., 34 complemented, 111 Weyl’s lemma, 165 orthogonal, 47 Whitney’s notation, 30 topological, 3 Wintner, A., 272 Sufficient family of semi-norms, 76 Wronskian, 65 Summability kernel, 54, 56 Wronskian determinant, 64 Support of a distribution, 155 Young’s inequality, 54, 63 of a function, 7 Young, W. H., 10, 14 of a spectral measure, 246 Sylvester, J., 114 Zorn’s lemma, 6, 50, 107 Symmetry, 150, 191

Taylor, A., 232 Thorin, O., 59 Three lines theorem, 59 Topological vector space, 26 Topologically complementary subspaces, 111 Topology, 2 coarser, 3 finer, 3 metrizable, 4 of a metric space, 3 product, 6 weak, 129 weak*, 130 Total subset, 36 Translation, 150, 191 Transpose, 96, 113 Transposition map, 114 Functional analysis studies the algebraic, geometric, and topological 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 determined 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.

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