sign in sign up
<<
Home , Analytic semigroup, Fredholm operator, Functional analysis, Gelfand representation, Lp space, Minkowski functional

GRADUATE STUDIES IN 191

Functional

Theo Bühler Dietmar A. Salamon 10.1090/gsm/191

Functional Analysis

GRADUATE STUDIES IN MATHEMATICS 191

Functional Analysis

Theo Bühler Dietmar A. Salamon EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky

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

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

Library of Congress Cataloging-in-Publication Data Names: B¨uhler, Theo, 1978– author. | Salamon, D. (Dietmar), author. Title: Functional analysis / Theo B¨uhler, Dietmar A. Salamon. Description: Providence, Rhode Island : American Mathematical Society, [2018] | : Gradu- ate studies in mathematics ; volume 191 | Includes bibliographical references and index. Identifiers: LCCN 2017057159 | ISBN 9781470441906 (alk. paper) Subjects: LCSH: Functional analysis. | (Mathematics) | of operators. | AMS: Functional analysis – Instructional exposition (textbooks, tutorial papers, etc.). msc | theory – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA320 .B84 2018 | DDC 515/.7—dc23 LC record available at https://lccn.loc.gov/2017057159

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 Contents

Preface ix Introduction xi

Chapter 1. Foundations 1 §1.1. Spaces and Compact Sets 2 §1.2. Finite-Dimensional Banach Spaces 17 §1.3. The Dual 25 §1.4. Hilbert Spaces 31 §1.5. Banach 35 §1.6. The 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* 109 §3.1. Topological Vector Spaces 110 §3.2. The Banach–Alaoglu Theorem 124 §3.3. The Banach–Dieudonn´e Theorem 130 §3.4. The Eberlein–Smulyanˇ Theorem 134

v vi Contents

§3.5. The Kre˘ın–Milman Theorem 140 §3.6. 144 §3.7. Problems 153 Chapter 4. 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 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 306 §6.3. Unbounded Operators on Hilbert Spaces 313 §6.4. 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 377 §7.4. Analytic Semigroups 388 §7.5. Valued Measurable Functions 404 §7.6. Inhomogeneous 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 the second author at ETH Z¨urich in the fall semester 2015. Prerequisites are the first year courses on Analysis and Linear , and the second year courses on , , and and Integration. The material of Section 1.4 on elementary theory, Sub- section 5.4.2 on the Stone–Weierstraß Theorem, and the appendix on the Lemma 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 the functional calculus for bounded normal operators, and Chapter 6 on un- bounded operators was not part of the lecture course (with the exception of some of the basic definitions in Chapter 6 that are relevant for infinitesimal generators of strongly continuous semigroups). From Chapter 7 only the basic material on strongly continuous semigroups in Section 7.1, on their infinitesimal 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 B¨uhler Dietmar A. Salamon

ix

Introduction

Classically, functional analysis is the study of spaces and linear operators between them. The relevant function spaces are often equipped with the structure of a Banach space and many of the central results re- main valid in the more general setting of bounded linear operators between Banach spaces or normed vector spaces, where the specific properties of the concrete in question only play a minor role. Thus, in the modern guise, functional analysis is the study of Banach spaces and bounded linear operators between them, and this is the viewpoint taken in the present book. This area of mathematics has both an intrinsic beauty, which we hope to convey to the reader, and a vast number of applications in many fields of mathematics. These include the analysis of PDEs, differential topology and , symplectic topology, , theory, geo- metric , dynamical systems, ergodic theory, and , among many others. While we say little about specific applications, they do motivate the choice of topics covered in this book, and our goal is to give a self-contained exposition of the necessary background in abstract functional analysis for many of the relevant applications. The book is addressed primarily to third year students of mathematics or physics, and the reader is assumed to be familiar with first year analysis and , as well as complex analysis and the basics of point topology and measure and integration. For example, this book does not include a proof of completeness and for Lp spaces. There are naturally many topics that go beyond the scope of the present book, such as Sobolev spaces and PDEs, which would require a book on its own 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 functional analysis. Other topics not covered include the (see [2,48, 79]), maximal regularity for semigroups (see [76]), the space of Fredholm operators on an infinite-dimensional Hilbert space as a classifying space for K-theory (see [5–7,42]), Quillen’s line bundle over the space of Fredholm operators (see [71,77]), and the work of Gowers [31] and Argyros– Haydon [4] on Banach spaces on which every bounded linear operator is the sum of a scalar multiple of the identity and a . Here is a description of the contents of the book, chapter by chapter. Chapter 1 discusses some basic concepts that play a central role in the subject. It begins with a section on metric spaces and compact sets which includes a proof of the Arzel`a–Ascoli Theorem. It then moves on to establish some basic properties of finite-dimensional normed vector spaces and shows, in particular, that a normed is finite-dimensional if and only if the unit ball is compact. The first chapter also introduces the of a , explains several important examples, and contains an introduction to elementary Hilbert space theory. It then introduces Banach algebras and shows that the group of invertible elements is an . It closes with a proof of the Baire Category Theorem. Chapter 2 is devoted to the three fundamental principles of functional analysis.TheyaretheUniform Boundedness Principle (a pointwise bounded family of bounded linear operators on a Banach space is bounded), the Open Mapping Theorem (a surjective bounded linear operator between Banach spaces is open), and the Hahn–Banach Theorem (a bounded linear func- tional on a of a normed vector space extends to a bounded linear functional on the entire normed vector space). An equivalent formu- lation of the Open Mapping Theorem is the Theorem (a linear operator between Banach spaces is bounded if and only if it has a closed graph) and a corollary is the Inverse Operator Theorem (a bijective bounded linear operator between Banach spaces has a bounded inverse). A slightly stronger version of the Hahn–Banach Theorem, with the replaced by a quasi-, can be reformulated as the geometric assertion that two convex subsets of a normed vector space can be separated by a hyperplane whenever one of them has nonempty . The chapter also discusses reflexive Banach spaces and includes an exposition of the James space. The subjects of Chapter 3 are the on a Banach space X and the weak* topology on its dual space X∗. With these topologies X and X∗ are locally convex Hausdorff topological vector spaces and the chap- ter begins with a discussion of the elementary properties of such spaces. The central result of the third chapter is the Banach–Alaoglu Theorem which asserts 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 of mathematics. The chapter also contains a proof of the Banach–Dieudonn´e Theorem which asserts that a linear subspace of the dual space of a Banach space is weak* closed if and only if its intersection with the closed unit ball is weak* closed. A consequence of the Banach–Alaoglu Theorem is that the unit ball in a reflexive Banach space is weakly compact, and the Eberlein– Smulyanˇ Theorem asserts that this property characterizes reflexive Banach spaces. The Kre˘ın–Milman Theorem asserts that every nonempty compact convex subset of a locally convex Hausdorff is the closed of its extremal points. Combining this with the Banach– Alaoglu Theorem, one can prove that every homeomorphism of a compact admits an invariant ergodic Borel probability measure. Some properties 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 Fredholm operators and their indices including the stability theorem. A is a bounded linear operator between Banach spaces that has a finite-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 index form an open subset of the space of all bounded linear operators between two Banach spaces, with respect to the topology induced by the . It also asserts that the sum of a Fredholm operator and a compact operator is again Fredholm and has the same index as the original operator. The chapter includes an introduction to the dual of a bounded linear operator, a proof of the Closed Image Theorem which asserts that an operator has a closed image if and only if its dual does, an introduction to compact operators which map the unit ball to pre-compact subsets of the target space, a characterization of 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 linear operator on a real or complex Banach space. A first preparatory section discusses complex Banach spaces and the complexifications of real Banach spaces, the of continuous Banach space valued functions on com- pact intervals, and holomorphic operator valued functions. The chapter then introduces the spectrum of a bounded linear operator, examines its elemen- tary properties, discusses the spectra of compact operators, and establishes the holomorphic functional calculus. The remainder of this chapter deals exclusively with operators on Hilbert spaces, starting with a discussion of complex Hilbert spaces and the spectra of normal and self-adjoint operators. It then moves on to C* algebras and the continuous functional calculus for self-adjoint operators, which takes the form of an from the xiv Introduction

C* algebra of complex valued continuous functions on the spectrum to the smallest C* algebra containing the given operator. The next topic is the and the extension of the continuous functional cal- culus to normal operators. The chapter also contains a proof that every can be represented by a projection valued measure on the spectrum, and that every self-adjoint operator is isomorphic to a direct sum of 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 linear subspace. 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 an unbounded operator and an extension of the Closed Image Theorem to this setting. It then examines the basic properties of the spectra of unbounded operators. The remainder of the chapter deals with unbounded operators on Hilbert spaces and their adjoints. In particular, it extends the functional calculus 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 partial differential equations such as the heat , the wave equation, and the Schr¨odinger equation, and they can be viewed as infinite-dimensional ana- logues of the exponential S(t):=etA. In all the relevant examples the operator A is unbounded. It is called the infinitesimal generator of the strongly continuous semigroup in question. A central result in the subject is the Hille–Yosida–Phillips Theorem which characterizes the infinitesimal generators of strongly continuous semigroups. The dual semigroup is not always strongly continuous. It is, however, strongly continuous whenever the Banach space in question is reflexive. The chapter also includes a basic treatment of analytic semigroups and their infinitesimal generators. It closes with a study of Banach space valued measurable functions and of solutions to the inhomogeneous equation associated to a semigroup. Each of the seven chapters ends with a problem section, which we hope will give the interested reader the opportunity to deepen their understanding of the subject.

Bibliography

[1]L.V.Ahlfors,Complex analysis: An introduction to the theory of analytic functions of one complex variable, 3rd ed., International Series in Pure and , McGraw- Hill Book Co., New York, 1978. MR510197 [2] N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space. Vols. I & II, Translated from the third Russian edition byE.R.Dawson;TranslationeditedbyW.N. Everitt, Monographs and Studies in Mathematics, Vols. 9 & 10, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR615737 [3] F. Albiac and N. J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR2192298

[4] S. A. Argyros and R. G. Haydon, A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem,ActaMath.206 (2011), no. 1, 1–54. MR2784662 [5] M. F. Atiyah, and operators in Hilbert space, Lectures in Modern Analysis and Applications. I, Springer, Berlin, 1969, pp. 101–121. MR0248803 [6]M.F.Atiyah,Bott periodicity and the index of elliptic operators,Quart.J.Math.Oxford Ser. (2) 19 (1968), 113–140. MR0228000 [7] M. F. Atiyah, K-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR0224083 [8] S. Banach, Th´eorie des op´erations lin´eaires, Monografje Matewatyczne, Warsaw, 1932. [9] A. Benedek, A.-P. Calder´on, and R. Panzone, operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365. MR0133653 [10] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1,Ameri- can Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR1727673 [11] O. V. Besov, On some families of functional spaces. Imbedding and extension theorems (Russian), Dokl. Akad. Nauk SSSR 126 (1959), 1163–1165. MR0107165 [12] O. V. Besov, Equivalent normings of spaces of functions of variable smoothness (Russian, with Russian summary), Tr. Mat. Inst. Steklova 243 (2003), 87–95; English transl., Proc. Steklov Inst. Math. 4(243) (2003), 80–88. MR2049464 [13] G. D. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. U.S.A. 17 (1931), 656–660. [14] G. D. Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), no. 2, 357–378. MR1501815

453 454 Bibliography

[15] Salomon Bochner, Integration von Funkionen, deren Werte die Elemente eines * Vector- raumes sind, Fund. Math. 20 (1933), 262–276. * * [16] S. Bochner, Absolut-additive abstrakte Mengenfunctionen, Fund. Math. 21 (1933), 211–213. [17] S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly-valued func- tions,Ann.ofMath.(2)39 (1938), no. 4, 913–944. MR1503445 [18] B. Bockelman, Analysis notes: The Stone–Weierstraß Theorem, October 2005. http://www.math.unl.edu/~s-bbockel1/922-notes/Stone_Weierstrass_Theorem.html [19] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations,Universitext, Springer, New York, 2011. MR2759829 [20] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414. MR1501880 [21] J. Diestel and J. J. Uhl Jr., The Radon-Nikod´ym theorem for Banach space valued measures, Rocky Mountain J. Math. 6 (1976), no. 1, 1–46. MR0399852 [22] J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Math- ematical Society, Providence, R.I., 1977. MR0453964 [23] R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. MR0203464 [24] N. Dunford and A. P. Morse, Remarks on the preceding paper of James A. Clarkson: “Uni- formly convex spaces” [Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414; MR1501880, Trans. Amer. Math. Soc. 40 (1936), no. 3, 415–420. MR1501881 [25] N. Dunford and B. J. Pettis, Linear operations on summable functions,Trans.Amer.Math. Soc. 47 (1940), 323–392. MR0002020 [26] N. Dunford and J. T. Schwartz, Linear operators. Part I: General theory, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. [27] P. Enflo, A counterexample to the approximation problem in Banach spaces,ActaMath.130 (1973), 309–317. MR0402468 [28] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR0181836 [29] I. M. Gelfand, Abstrakte Functionen und lineare Operatoren, Math. Sbornik N.S. 46 (1938), 235–286. [30] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,2nded., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR737190 [31] W. T. Gowers, A solution to Banach’s hyperplane problem, Bull. London Math. Soc. 26 (1994), no. 6, 523–530. MR1315601 [32] A. Grigor’yan and L. Liu, and Lipschitz-Besov spaces,ForumMath.27 (2015), no. 6, 3567–3613. MR3420351 [33] A. Grothendieck, Produits tensoriels topologiques et espaces nucl´eaires (French), Mem. Amer. Math. Soc. No. 16 (1955), 140. MR0075539 [34] H. Herrlich, , Lecture Notes in Mathematics, vol. 1876, Springer-Verlag, Berlin, 2006. MR2243715 [35] E. Hille, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, New York, 1948. MR0025077 [36] R. B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Math- ematics, No. 24, Springer-Verlag, New York-Heidelberg, 1975. MR0410335 [37] R. C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518–527. MR0039915 [38] R. C. James, A non-reflexive Banach space isometric with its second conjugate space,Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174–177. MR0044024 Bibliography 455

[39] R. C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140. MR0165344 [40] R. C. James, Characterizations of reflexivity, Studia Math. 23 (1963/1964), 205–216. MR0170192 [41] R. C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738–743. MR0417763 [42] K. J¨anich, Vektorraumb¨undel und der Raum der Fredholm-Operatoren, Math. Ann. 161 (1965), 129–142. MR0190946 [43] K. Jarosz, Any Banach space has an equivalent norm with trivial , Israel J. Math. 64 (1988), no. 1, 49–56. MR981748 [44] T. Kato, for linear operators, Die Grundlehren der mathematischen Wis- senschaften, 132, Corrected printing of the second edition, Springer-Verlag New York, Inc., New York, 1980. MR0203473 [45] J. L. Kelley, , Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.] MR0370454 [46] G. Kerkyacharian and P. Petrushev, Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces,Trans.Amer.Math.Soc.367 (2015), no. 1, 121–189. MR3271256 [47] M. Krein and D. Milman, On extreme points of regular convex sets (English., with Ukrainian summary), Studia Math. 9 (1940), 133–138. MR0004990 [48] A. W. Knapp, Basic , Cornerstones, Birkh¨auser Boston, Inc., Boston, MA, 2005. Digital second edition 2016. MR2155259 [49] A. W. Knapp, Advanced real analysis, Cornerstones, Birkh¨auser Boston, Inc., Boston, MA, 2005. Digital second edition 2017. MR2155260 [50] M. Kreuter, Sobolev spaces of vector valued functions, Master Thesis, Universit¨at Ulm, 2015. [51] R. Krom, The Donaldson geometric flow is a local smooth semiflow, Preprint, December 2015. http://arxiv.org/abs/1512.09199 [52] R. Krom, D. A. Salamon, The Donaldson geometric flow for symplectic four-, Preprint, December 2015. http://arxiv.org/abs/1512.09198 [53] N. H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19–30. MR0179792 [54] D. Lamberton, Equations´ d’´evolution lin´eaires associ´ees `a des semi-groupes de contractions dans les espaces Lp (French), J. Funct. Anal. 72 (1987), no. 2, 252–262. MR886813 [55] P. D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley- Interscience [John Wiley & Sons], New York, 2002. MR1892228 [56] I. Leader, Logic and , University of Cambridge, Michaelmas 2003. [57] L. Liu, D. Yang, and W. Yuan, Besov-type and Triebel-Lizorkin-type spaces associated with heat kernels,Collect.Math.67 (2016), no. 2, 247–310. MR3484021 [58] G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679–698. MR0132403 [59] R. E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998. MR1650235 [60] D. Milman, Characteristics of extremal points of regularly convex sets (Russian), Doklady Akad. Nauk SSSR (N.S.) 57 (1947), 119–122. MR0022313 [61] J. R. Munkres, Topology: a first course, second edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 2000. MR0464128 [62] J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Natl. Acad. Sci. U.S.A. 18 (1932), 70–82. [63] J. Peetre, On spaces of Triebel-Lizorkin type,Ark.Mat.13 (1975), 123–130. MR0380394 456 Bibliography

[64] J. Peetre, New thoughts on Besov spaces, Mathematics Department, Duke University, Durham, N.C., 1976. Duke University Mathematics Series, No. 1. MR0461123 [65] R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516–541. MR0004094 [66] R. S. Phillips, On weakly compact subsets of a Banach space,Amer.J.Math.65 (1943), 108–136. MR0007938 [67] R. S. Phillips, On the generation of semigroups of linear operators, Pacific J. Math. 2 (1952), 343–369. MR0050797 [68] R. S. Phillips, A note on the abstract Cauchy problem, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 244–248. MR0061759 [69] R. S. Phillips, The adjoint semi-group, Pacific J. Math. 5 (1955), 269–283. MR0070976 [70] J. D. Pryce, Weak compactness in locally convex spaces, Proc. Amer. Math. Soc. 17 (1966), 148–155. MR0190695 [71] D. Quillen, of Cauchy-Riemann operators on Riemann surfaces (Russian), Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41, 96; English transl., Funct. Anal. Appl. 19 (1985), 31–34. MR783704 [72] M. Reed, B. Simon, Methods of modern I: Functional analysis, Revised and Enlarged Edition, Academic Press, 1980. [73] W. Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathemat- ics, McGraw-Hill, Inc., New York, 1991. MR1157815 [74] D. A. Salamon, Funktionentheorie,Birkh¨auser, 2012. http://www.math.ethz.ch/~salamon/PREPRINTS/cxana.pdf [75] D. A. Salamon, Measure and integration, EMS Textbooks in Mathematics, European Math- ematical Society (EMS), Z¨urich, 2016. MR3469972 [76] D. A. Salamon, Parabolic Lp − Lq estimates, ETH Preprint, 2016. http://www.math.ethz.ch/~salamon/PREPRINTS/parabolic.pdf [77] D. A. Salamon, Notes on the universal determinant bundle, ETH, Preprint, 2013. http://www.math.ethz.ch/~salamon/PREPRINTS/det.pdf [78] L. Schwartz, Functional analysis, Notes by Paul Rabinowitz. Courant Institute, NYU, 1964. [79] B. Simon, A Comprehensive Course in Analysis, five vols, American Mathematical Society, Providence, RI, 2015. MR3364090 [80] A. D. Sokal, A really simple elementary proof of the uniform boundedness theorem,Amer. Math. Monthly 118 (2011), no. 5, 450–452. MR2805031 [81] M. H. Stone, On one-parameter unitary groups in Hilbert space, Ann. of Math. (2) 33 (1932), no. 3, 643–648. MR1503079 [82] A. Szankowski, Subspaces without the ,IsraelJ.Math.30 (1978), no. 1-2, 123–129. MR508257 [83] A. Szankowski, B(H) does not have the approximation property,ActaMath.147 (1981), no. 1-2, 89–108. MR631090 [84] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Math- ematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR503903 [85] H. Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkh¨auser Verlag, Basel, 1992. MR1163193 [86] D. Werner, Funktionalanalysis, Springer-Verlag, Berlin, 2004. MR1787146 [87] K. Yosida, On the differentiability and the representation of one-parameter semi-group of linear operators, J. Math. Soc. Japan 1 (1948), 15–21. MR0028537 Bibliography 457

[88] K. Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR617913

Notation

2X , set of all subsets of a set X,2 A, , 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 ,Borelσ-algebra, 30 c0, space of 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), , 208 (bounded), 299 (unbounded) −1 Rλ(A)=(λ1l − A) , 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⊥, , 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), 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) 2,2 ∩ 1,2 ⊂ Rn W (Ω) W0 (Ω), Sobolev space and on Ω , 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 reflexive, 81, 134 adjoint operator, 165 separable, 85 complex, 225 strictly convex, 143 unbounded, 313 Banach–Alaoglu Theorem affine hyperplane, 72 general case, 126, 155 Alaoglu–Bourbaki Theorem, 156 separable case, 124 almost everywhere, 4 Banach–Dieudonn´e Theorem, 130 annihilator, 74 Banach–Mazur Theorem, 158 left, 120 Banach–Steinhaus Theorem, 52 approximation property, 178 Argyros–Haydon space, 188 orthonormal, 79 Arzel`a–Ascoli Theorem, 13, 15, 16, 155 Schauder, 99, 106, 178 Atiyah–J¨anich Theorem, 188 Bell–Fremlin Theorem, 162 axiom of choice, 446 bidual Bell–Fremlin Theorem, 162 operator, 165 axiom of countable choice, 6 space, 80 axiom of dependent choice, 6, 48 continuous, 53 Babylonian method positive definite, 31 for square roots, 236 symmetric, 31 Baire Category Theorem, 42 symplectic, 314, 345 Banach algebra, 35, 234 Birkhoff’s Ergodic Theorem, 145 ideal, 246 Birkhoff–von Neumann Theorem, 160 semisimple, 246 Borel σ-algebra, 30 Banach hyperplane problem, 188 Borel measurable operator, 48 Banach , 104 bounded Banach space, 3 , 53 approximation property, 178 linear operator, 17 complex, 100, 198 invertible, 39 complexified, 201 pointwise, 50 product, 25 Bourbaki–Witt Theorem, 446 quotient, 24 Radon–Nikod´ym property, 415 C* algebra, 234

461 462 Index

Calkin algebra, 187 face, 140 Cantor function, 158 separation, 70, 116, 123 category cyclic vector, 281 in the sense of Baire, 40 Cauchy formula, 207 deformation retract, 193 Cauchy problem, 349, 363 dense well-posed, 363 linear subspace, 76 Cauchy , 2 subset, 11 Cauchy–Schwarz inequality, 31 direct sum, 57 complex, 222 Dirichlet problem, 320 Cayley transform, 327 , 374 Cayley–Hamilton Theorem, 291 doubly stochastic matrix, 160 chain, 445 dual operator, 164, 306 closeable linear operator, 62 complex, 198 closed convex hull, 120 dual space, 26 , 59 complex, 198 1 Closed Image Theorem, 169, 308 of  ,29 p closed linear operator, 59 of  ,28 cokernel, 179 of C(M), 30 of c0,29 comeagre, 40 p compact of L (μ), 26 subset of a , 5 of a Hilbert space, 32 finite intersection property, 450 of a quotient, 76 of a subspace, 76 operator, 173–177 Dunford integral, 216, 305 pointwise, 12 subset of a metric space, 5 Eberlein–Smulyanˇ Theorem, 134 compact-open topology, 154 eigenspace , 78 generalized, 213 complete eigenvalue, 208, 299 metric space, 3 eigenvector, 208, 299 subset of a metric space, 5 equi-continuous, 12, 16 completely continuous operator, 174 equivalent norms, 18 completion of a metric space, 45 ergodic complexification measure, 144 of a linear operator, 201 theorem, 148 of a norm, 201 Birkhoff, 145 of a vector space, 201 von Neumann, 146 of the dual space, 202 uniquely, 145 exact sequence, 195 vanishing at infinity, 129, 155 Euler characteristic, 195 weakly, 404 extremal point, 140 contraction semigroup, 374 convergence Fej´er’s Theorem, 79 in measure, 111 finite intersection property, 450 weak, 114 first category, 40 weak*, 114 flow, 352 convex hull, 120 formal adjoint of a differential operator, 64 absorbing, 104 , 79, 103 and interior, 73, 115 Fredholm extremal point, 140 alternative, 187, 193 Index 463

index, 179 image, 18 operator, 179 infinitesimal generator, 357 Stability Theorem, 185 of a contraction semigroup, 375 triple, 194 of a group, 366 functional calculus of a self-adjoint semigroup, 382 bounded measurable, 267 of a shift group, 385 continuous, 240, 257 of a unitary group, 384 holomorphic, 217, 305 of an , 393 normal, 257, 267 of the dual semigroup, 377 self-adjoint, 240, 326 of the heat semigroup, 403 unbounded, 326 Schr¨odinger operator, 386 uniqueness of the semigroup, 365 Gantmacher’s Theorem, 190 well-posed Cauchy problem, 363 Gelfand representation, 249 inner product, 31 Gelfand spectrum, 246, 293 Hermitian, 79, 222 2 Gelfand transform, 249, 293 on L (μ), 34 Gelfand triple, 316 integral Gelfand–Mazur Theorem, 248, 290 Banach space valued, 203, 410 Gelfand–Robbin quotient, 315, 346 value inequality, 204 graph norm, 59, 296, 361 over a curve, 205 invariant measure, 125 Hahn–Banach Theorem, 65 ergodic, 144 closure of a subspace, 74 inverse in a Banach algebra, 35 for bounded linear functionals, 67 inverse operator, 39 for convex sets, 70, 116, 123 Inverse Operator Theorem, 56 for positive linear functionals, 68 , 246 , 239 James’ space, 86–100 heat James’ Theorem, 134 equation, 352, 403 joint kernel, 120 kernel, 352, 403 Hellinger–Toeplitz Theorem, 61 K-theory, 188 Helly’s Theorem, 135, 158 kernel, 18, 179 Hermitian inner product, 222 Kre˘ın–Milman Theorem, 140 2 N C on  ( , ), 224 Kronecker symbol, 28 2 C on L (μ, ), 224 Kuiper’s Theorem, 188 on L2(R/Z, C), 79 Hilbert cube, 143, 161 Lagrangian subspace, 314, 345 Hilbertspace,31 Laplace operator, 298, 320 complex, 223 linear functional complexification, 223 bounded, 17 dual space, 26 positive, 68 , 79 linear operator separable, 79 adjoint, 165, 225 unit sphere contractible, 193 bidual, 165 Hille–Yosida–Phillips Theorem, 368–374 bounded, 17 H¨older inequality, 26 closeable, 62 holomorphic closed, 59 function, 205 cokernel, 179 functional calculus, 216–221 compact, 174, 233 hyperplane, 72 completely continuous, 174 affine, 72 complexified, 201 464 Index

cyclic vector, 281 spectral, 263 dissipative, 374 metric space, 2 dual, 164 compact, 5 exponential map, 221 complete, 3 finite rank, 174 completion, 45 Fredholm, 179 Milman–Pettis Theorem, 156 image, 179 , 104 inverse, 39 kernel, 179 nonmeagre, 40 logarithm, 221 norm, 3 normal, 227, 321 equivalent, 18 positive semidefinite, 245 operator, 17 projection, 78, 147 normal operator, 227 right inverse, 78 spectrum, 229 self-adjoint, 165, 227, 313 unbounded, 321 , 233 normed vector space, 3 spectrum, 208 dual space, 26 , 221, 245 weak* topology, 114 symmetric, 61, 63 product, 24 unitary, 227 quotient, 23 weakly compact, 190 strictly convex, 106 linear subspace uniformly convex, 156 closure, 76 weak topology, 114 complemented, 78 nowhere dense, 40 dense, 76 dual of, 76 open invariant, 288 ball, 2 orthogonal complement, 225 half-space, 72 weak* closed, 130 map, 54 weak* dense, 122 set in a metric space, 2 weakly closed, 119 Open Mapping Theorem, 54 Lipschitz continuous, 413 for unbounded operators, 343 long exact sequence, 195 operator norm, 17 Lumer–Phillips Theorem, 375 , 68 orthogonal complement, 34 Markov–Kakutani complex, 225 Fixed Point Theorem, 161 orthonormal basis, 79 maximal ideal, 246 meagre, 40 partial order, 445 Mean Ergodic Theorem, 145 Pettis’ Lemma, 48 Pettis’ Theorem, 405 Banach space valued, 404 Phillips’ Lemma, 101 strongly, 404 Pitt’s Theorem, 191 weakly, 404 pointwise measure bounded, 50 complex, 200 compact, 12 ergodic, 144 precompact, 12 invariant, 125 , 156 probability, 125 positive cone, 68 projection valued, 262 positive linear functional, 68 pushforward, 165 pre-annihilator, 120 signed, 4 precompact Index 465

pointwise, 12 radius, 37, 39, 211 subset of a metric space, 5 Spectral Mapping Theorem subset of a topological space, 5 bounded linear operators, 217 probability measure, 125 normal operators, 267 product space, 25 self-adjoint operators, 240, 326 product topology, 110, 113, 450 unbounded operators, 326 projection, 78, 147 , 281 spectrum, 208 quasi-seminorm, 65 continuous, 208, 299 quotient space, 23 in a unital Banach algebra, 246 dual of, 76 of a , 246 of a compact operator, 213 , 155 of a normal operator, 229, 324 Radon–Nikod´ym property, 415 of a self-adjoint operator, 231 reflexive Banach space, 80–84, 134 of a , 229 residual, 40 of an unbounded operator, 299 resolvent point, 208, 299 identity, 210, 300 residual, 208, 299 for semigroups, 368 square root, 245, 344 operator, 210, 299 Babylonian method, 236 set, 208, 299 Stone’s Theorem, 384 Riemann–Lebesgue Lemma, 103 Stone–Weierstraß Theorem, 236, 289, Riesz Lemma, 22 290 Ruston’s Theorem, 106 strictly convex, 106, 143, 160 strong convergence, 52 Schatten’s tensor product, 105 strongly continuous semigroup, 350 , 99, 106, 178 analytic, 388–403 Schr¨odinger equation, 386 contraction, 374 Schr¨odinger operator, 298 dual semigroup, 377 Schur’s Theorem, 121, 153 extension to a group, 366 second category, 40 heat kernel, 352, 403 self-adjoint operator, 165, 227 Hille–Yosida–Phillips, 368 spectrum, 231 infinitesimal generator, 357 unbounded, 313 on a Hilbert space, 351 semigroup regularity problem, 429 strongly continuous, 350 Schr¨odinger equation, 386 seminorm, 65 self-adjoint, 382 separable shift operators, 351, 385 Banach space, 85 unitary group, 384 Hilbert space, 79 well-posed Cauchy problem, 363 topological space, 11 symmetric linear operator, 61, 165 signed measure, 4 symplectic total variation, 333 form, 314, 345 simplex reduction, 345 infinite-dimensional, 143 vector space, 314, 345 singular value, 233 Smulyan–Jamesˇ Theorem, 159 tensor product, 105 Snake Lemma, 195 topological vector space, 110 Sobolev space, 423 locally convex, 110 spectral topology, 2 measure, 263, 332 compact-open, 154 projection, 215, 305 of a metric space, 2 466 Index

of a normed vector space, 3 product, 110, 113, 450 strong, 110, 114 strong operator, 52 uniform operator, 17 weak, 114 weak*, 114 total variation of a signed measure, 333 totally bounded, 5 , 2, 31 trigonometric polynomial, 290 Tychonoff’s Theorem, 450 unbounded operator, 295 densely defined, 295 normal, 321 self-adjoint, 313 spectral projection, 305 spectrum, 299, 324 with compact resolvent, 302 Uniform Boundedness Theorem, 50 unitary operator, 227 spectrum, 229 vector space complex normed, 198 complexification, 201 normed, 3 ordered, 68 topological, 110 , 291 von Neumann’s Mean Ergodic Theorem, 146 wave equation, 353, 387 weak compactness, 134–139 continuity, 404 convergence, 114 measurability, 404 topology, 114, 119–121 weak* compactness, 126, 127 convergence, 114 sequential closedness, 127 sequential compactness, 124, 127 topology, 114, 122–123, 130–133 weakly compact, 190 winding number, 216

Zorn’s Lemma, 446 Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction XSXLI½IPHJSVKVEHYEXIWXYHIRXWERHVIWIEVGLIVW 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 HMWGYWWIWVI¾I\MZIWTEGIWERHXLI.EQIWWTEGI'LETXIVMRXVSHYGIWXLI[IEOERH weak∗ topologies and includes the theorems of Banach–Alaoglu, Banach–Dieudonné, Eberlein–SÛmulyan, 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 FSYRHIH PMRIEV STIVEXSVW -X MRXVSHYGIW GSQTPI\ &EREGL ERH ,MPFIVX WTEGIW XLI 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 XLISVIQMRXLMWWIXXMRKERHI\XIRHWXLIJYRGXMSREPGEPGYPYWERHWTIGXVEPQIEWYVIXS unbounded self-adjoint operators on Hilbert spaces. Chapter 7 gives an introduction XSWXVSRKP]GSRXMRYSYWWIQMKVSYTWERHXLIMVMR½RMXIWMQEPKIRIVEXSVW-XMRGPYHIWJSYR- HEXMSREPVIWYPXWEFSYXXLIHYEPWIQMKVSYTERHEREP]XMGWIQMKVSYTWERI\TSWMXMSRSJ measurable functions with values in a Banach space, and a discussion of solutions to XLIMRLSQSKIRISYWIUYEXMSRERHXLIMVVIKYPEVMX]TVSTIVXMIW8LIETTIRHM\IWXEFPMWLIW XLIIUYMZEPIRGISJXLI0IQQESJ>SVRERHXLI%\MSQSJ'LSMGIERHMXGSRXEMRWE proof of Tychonoff’s theorem. ;MXLXSIPEFSVEXII\IVGMWIWEXXLIIRHSJIEGLGLETXIVXLMWFSSOGERFIYWIH EWEXI\XJSVESRISVX[SWIQIWXIVGSYVWISRJYRGXMSREPEREP]WMWJSVFIKMRRMRKKVEH- YEXIWXYHIRXW4VIVIUYMWMXIWEVI½VWX]IEVEREP]WMWERHPMRIEVEPKIFVEEW[IPPEWWSQI JSYRHEXMSREPQEXIVMEPJVSQXLIWIGSRH]IEVGSYVWIWSRTSMRXWIXXSTSPSK]GSQTPI\ analysis in one variable, and measure and integration.

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

GSM/191 www.ams.org