Springer Monographs in Mathematics V.I. Bogachev O.G. Smolyanov Topological Vector Spaces and Their Applications Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research. More information about this series at http://www.springer.com/series/3733 V.I. Bogachev · O.G. Smolyanov Topological Vector Spaces and Their Applications 1 3 V.I. Bogachev O.G. Smolyanov Department of Mechanics Department of Mechanics and Mathematics and Mathematics Moscow State University Moscow State University Moscow Moscow Russia Russia and Higher School of Economics National Research University Moscow Russia ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-319-57116-4 ISBN 978-3-319-57117-1 (eBook) DOI 10.1007/978-3-319-57117-1 Library of Congress Control Number: 2017939903 Mathematics Subject Classification: 46A03, 58C20, 28C20 The book is an expanded and revised version of the Russian edition under the same title, published in Regular and Chaotic Dynamics, Moscow-Izhevsk, 2012. © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microflms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Contents Preface ix Chapter 1. Introduction to the theory of topological vector spaces 1 1.1. Linear spaces and topology 1 1.2. Basic definitions 11 1.3. Examples 17 1.4. Convex sets 29 1.5. Finite-dimensional and normable spaces 35 1.6. Metrizability 41 1.7. Completeness and completions 45 1.8. Compact and precompact sets 53 1.9. Linear operators 59 1.10. The Hahn–Banach theorem: geometric form 63 1.11. The Hahn–Banach theorem: analytic form 71 1.12. Complements and exercises 81 Uniform spaces (81). Convex compact sets (84). Fixed point theorems (86). Sequence spaces (89). Duals to Banach spaces (90). Separability properties (91). Continuous selections and extensions (93). Exercises (94). Chapter 2. Methods of constructing topological vector spaces 101 2.1. Projective topologies 101 2.2. Examples of projective limits 104 2.3. Inductive topologies 109 2.4. Examples of inductive limits 113 2.5. Grothendieck’s construction 119 2.6. Strict inductive limits 125 2.7. Inductive limits with compact embeddings 127 2.8. Tensor products 130 2.9. Nuclear spaces 134 2.10. Complements and exercises 139 Properties of the spaces D and D (139). Absolutely summing operators (143). Local completeness (145). Exercises (147). v vi Contents Chapter 3. Duality 153 3.1. Polars 153 3.2. Topologies compatible with duality 158 3.3. Adjoint operators 162 3.4. Weak compactness 164 3.5. Barrelled spaces 170 3.6. Bornological spaces 175 3.7. The strong topology and reflexivity 180 3.8. Criteria for completeness 186 3.9. The closed graph theorem 193 3.10. Compact operators 199 3.11. The Fredholm alternative 205 3.12. Complements and exercises 208 Baire spaces (208). The Borel graph theorem (211). Bounding sets (212). The James theorem (213). Topological properties of locally convex spaces (214). Eberlein–Smulianˇ properties (218). Schauder bases (219). Minimal spaces and powers of the real line (221). Exercises (224). Chapter 4. Differential calculus 243 4.1. Differentiability with respect to systems of sets 244 4.2. Examples 251 4.3. Differentiability and continuity 257 4.4. Differentiability and continuity along a subspace 261 4.5. The derivative of a composition 263 4.6. The mean value theorem 273 4.7. Taylor’s formula 275 4.8. Partial derivatives 278 4.9. The inversion of Taylor’s formula and the chain rule 279 4.10. Complements and exercises 289 The inverse function theorem (289). Polynomials (291). Ordinary differential equations in locally convex spaces (294). Passage to the limit in derivatives (297). Completeness of spaces of smooth mappings (300). Differentiability via pseudotopologies (305). Smooth functions on Banach spaces (307). Exercises (308). Chapter 5. Measures on linear spaces 311 5.1. Cylindrical sets 311 5.2. Measures on topological spaces 313 5.3. Transformations and convergence of measures 321 5.4. Cylindrical measures 327 5.5. The Fourier transform 333 5.6. Covariance operators and means of measures 337 5.7. Gaussian measures 345 5.8. Quasi-measures 354 5.9. Sufficient topologies 357 5.10. The Sazonov and Gross–Sazonov topologies 359 Contents vii 5.11. Conditions for countable additivity 366 5.12. Complements and exercises 372 Convolution (372). 0–1 laws (376). Convex measures (378). The central limit theorem (381). Infinitely divisible and stable measures (383). Banach supports of measures (391). Infinite-dimensional Wiener processes (393). Prohorov locally convex spaces (394). Measurable linear and polylinear functions (399). Relations between different σ-algebras (409). Radonifying operators (412). Measurable norms (412). Exercises (413). Comments 419 Bibliography 425 Author index 447 Subject index 451 Preface This book gives a concise exposition of the fundamentals of the theory of topo- logical vector spaces, complemented by a survey of the most important results of a more subtle nature, which cannot be qualified as basic, but knowledge of which is useful for applications, and, finally, some of such applications connected with dif- ferential calculus in infinite-dimensional spaces and measure theory. Almost half of the book is devoted to these applications, which makes it very different from the whole series of known texts on topological vector spaces. Another notable dif- ference between this book and known treatises like Bourbaki [87], Edwards [150], Grothendieck [207], Jarchow [237], Kelley, Namioka [270], Kothe¨ [292], Nar- ici, Beckenstein [365], Perez´ Carreras, Bonet [385], Robertson, Robertson [420], Schaefer [436], Treves` [530], and Wilansky [567] is that we decided to include also some results without proofs (this does not concern the fundamentals, of course) with references instead, which enables us to inform the reader about many rela- tively recent achievements; some of them are disguised as exercises (with refer- ences to the literature), such exercises should not be confused with usual exercises marked by the symbol ◦. Thus, with respect to the presented information, our book is not covered by any other book on this subject (though, we cannot claim that it covers any such book). Chapter 1 contains the fundamentals of the theory, including a large list of concrete examples, some general concepts (convex sets, seminorms, linear map- pings) and a number of facts, the most important of which is the Hahn–Banach theorem on extensions of functionals in its diverse versions. The main material of Chapter 2 is connected with projective and inductive limits (including strict inductive limits and inductive limits with compact em- beddings, which is not sufficiently discussed in the existing literature), and also Grothendieck’s method of constructing Banach spaces embedded into locally con- vex spaces. Chapter 3 contains the classical material related to the so-called duality theory, i.e., introduction of different locally convex topologies on a given space giving the same set of continuous linear functionals. The central topics here are the Mackey– Arens theorem on topologies compatible with duality, the results on weak com- pactness, including the Eberlein–Smulianˇ and Krein–Smulianˇ theorems, and also some concepts and facts connected with completeness of locally convex spaces. ix x Preface Chapter 4 is devoted to the fundamentals of the differentiation theory in lo- cally convex spaces. It presents a general scheme of differentiability with respect to a system of sets (partial cases of which are Gateaux,ˆ Hadamard and Frechet´