Free ebooks ==> www.Ebook777.com Universitext Hans Wilhelm Alt Linear Functional Analysis An Application-Oriented Introduction Translated by Robert Nürnberg www.Ebook777.com Free ebooks ==> www.Ebook777.com Universitext www.Ebook777.com Universitext Series Editors Sheldon Axler San Francisco State University Vincenzo Capasso Università degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah CNRS, École Polytechnique, Paris Endre Süli University of Oxford Wojbor A Woyczy´nski Case Western Reserve University Cleveland, OH Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond The books, often well classtested by their author, may have an informal, personal even experimental approach to their subject matter Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext More information about this series at http://www.springer.com/series/223 Hans Wilhelm Alt Linear Functional Analysis An Application-Oriented Introduction Translated by Robert Nürnberg 123 Free ebooks ==> www.Ebook777.com Hans Wilhelm Alt Technische Universität München Garching near Munich Germany Translation from German language edition: Lineare Funktionalanalysis by Hans Wilhelm Alt Copyright © 2012, Springer Berlin Heidelberg Springer Berlin Heidelberg is part of Springer Science + Business Media All Rights Reserved ISSN 0172-5939 Universitext ISBN 978-1-4471-7279-6 DOI 10.1007/978-1-4471-7280-2 ISSN 2191-6675 (electronic) ISBN 978-1-4471-7280-2 (eBook) Library of Congress Control Number: 2016944464 Mathematics Subject Classification: 46N20, 46N40, 46F05, 47B06, 46G10 © Springer-Verlag London 1985, 1991, 1999, 2002, 2006, 2012, 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms 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 Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag London Ltd www.Ebook777.com Preface The present book is the English translation of a previous German edition, also published by Springer Verlag The translation was carried out by Robert Nă urnberg, who also did a marvellous job at detecting errors and mistakes in the original version In addition, Andrei Iacob revised the English version The book originated in a series of lectures I gave for the first time at the University of Bochum in 1980, and since then it has been repeatedly used in many lectures by me and other mathematicians and during this time it has changed accordingly I provide the reader with an introduction to Functional Analysis as a synthesis of Algebra, Topology, and Analysis, which is the source for basic definitions which are important for differential equations The book includes a number of appendices in which special subjects are presented in more detail Therefore its content is rich enough for a lecturer to find enough material to fill a course in functional analysis according to his special interests The text can also be used as an additional source for lectures on partial differential equations or advanced numerical analysis It must be said that my strategy has been dictated by the desire to offer the reader an easy and fast access to the main theorems of linear functional analysis and, at the same time, to provide complete proofs So there is a separate appendix where the Lebesgue integral is introduced in a complete functional analytic way, and an appendix whith details for Sobolev functions which complete the proofs of the embedding theorems Therefore the text is self-contained and the reader will benefit from this fact Parallel to this edition, a revised German version has become available (Lineare Funktionalanalysis, Edition, Springer 2012) with the same mathematical content This is made possible by a common source text Therefore one does not have to worry about the content in different versions I am happy that this book is now accessible to a wider community If you find any errors or misprints in the text, please point them out to the author via email: “alt@ma.tum.de” This will help to improve the text of possible future editions I hope that this book is written in the good tradition of functional analysis and will serve its readers well I thank Springer Verlag for making the publication of this edition possible and for their kind support over many years Technical University Munich, August 2015 H W Alt V Table of Contents Introduction Preliminaries 2.1 Scalar product 2.3 Orthogonality 2.4 Norm 2.6 Metric 2.8 Examples of metrics 2.9 Balls and distance between sets 2.10 Open and closed sets 2.11 Topology 2.14 Comparison of topologies 2.15 Comparison of norms 2.17 Convergence and continuity 2.18 Convergence in metric spaces 2.21 Completeness 2.22 Banach spaces and Hilbert spaces 2.23 Sequence spaces 2.24 Completion E2 Exercises E2.6 Completeness of Euclidean space E2.7 Incomplete function space E2.9 Hausdorff distance between sets 9 11 13 16 16 18 19 19 21 21 23 24 27 27 28 30 31 34 34 35 Function spaces 3.1 Bounded functions 3.2 Continuous functions on compact sets 3.3 Continuous functions 3.4 Support of a function 3.5 Differentiable functions 3.7 Hă older continuous functions 3.9 Measures 3.10 Examples of measures 3.11 Measurable functions 37 37 38 39 41 41 44 45 46 47 VII VIII Table of Contents 3.15 Lebesgue spaces 3.18 Hăolders inequality 3.19 Majorant criterion in Lp 3.20 Minkowski inequality 3.21 Fischer-Riesz theorem 3.23 Vitali’s convergence theorem 3.25 Lebesgue’s general convergence theorem 3.27 Sobolev spaces E3 Exercises E3.3 Standard test function E3.4 Lp -norm as p → ∞ E3.6 Fundamental theorem of calculus A3 Lebesgue’s integral A3.3 Elementary Lebesgue measure A3.4 Outer measure A3.5 Step functions A3.6 Elementary integral A3.8 Lebesgue integrable functions A3.10 Axioms of the Lebesgue integral A3.14 Integrable sets A3.15 Measure extension A3.18 Egorov’s theorem A3.19 Majorant criterion A3.20 Fatou’s lemma A3.21 Dominated convergence theorem 50 52 55 55 55 57 60 63 66 67 67 68 71 72 73 74 75 78 79 84 87 90 91 93 94 Subsets of function spaces 4.1 Convexity 4.3 Projection theorem 4.5 Almost orthogonal element 4.6 Compactness 4.12 Arzel`a-Ascoli theorem (compactness in C ) 4.13 Convolution 4.14 Dirac sequences 4.16 Riesz theorem (compactness in Lp ) 4.18 Examples of separable spaces 4.19 Cut-off function 4.20 Partition of unity 4.22 Fundamental lemma of calculus of variations 4.23 Local approximation of Sobolev functions 4.25 Product rule for Sobolev functions 4.26 Chain rule for Sobolev functions E4 Exercises E4.4 Strictly convex spaces E4.5 Separation theorem in IRn 95 95 96 99 100 106 107 110 112 115 118 118 122 122 124 125 126 128 129 Free ebooks ==> www.Ebook777.com Table of Contents IX E4.6 Convex functions E4.7 Characterization of convex functions E4.8 Supporting planes E4.9 Jensen’s inequality E4.11 The space Lp for p < E4.13 Compact sets in E4.15 Comparison of Hă older spaces E4.16 Compactness with respect to the Hausdorff metric E4.18 Continuous extension E4.19 Dini’s theorem E4.20 Nonapproximability in C 0,α E4.21 Compact sets in Lp 129 131 132 133 134 135 136 137 138 139 139 139 Linear operators 5.2 Linear operators 5.7 Neumann series 5.8 Theorem on invertible operators 5.9 Analytic functions of operators 5.10 Examples (exponential function) 5.12 Hilbert-Schmidt integral operators 5.14 Linear differential operators 5.17 Distributions (The space D (Ω)) 5.20 Topology on C0∞ (Ω) 5.21 The space D(Ω) E5 Exercises E5.3 Unique extension of linear maps E5.4 Limit of linear maps 141 142 146 147 147 148 149 151 152 156 157 160 160 161 Linear functionals 6.1 Riesz representation theorem 6.2 Lax-Milgram theorem 6.4 Elliptic boundary value problems 6.5 Weak boundary value problems 6.6 Existence theorem for the Neumann problem 6.7 Poincar´e inequality 6.8 Existence theorem for the Dirichlet problem 6.10 Variational measure 6.11 Radon-Nikod´ ym theorem 6.12 Dual space of Lp for p < ∞ 6.14 Hahn-Banach theorem 6.15 Hahn-Banach theorem (for linear functionals) 6.20 Spaces of additive measures 6.21 Spaces of regular measures 6.23 Riesz-Radon theorem 6.25 Functions of bounded variation 163 163 164 167 169 170 171 171 173 173 175 180 182 185 185 187 191 www.Ebook777.com ... Introduction Functional analysis deals with the structure of function spaces and the properties of continuous mappings between these spaces Linear functional analysis, in particular, is confined to the analysis. .. Unique extension of linear maps E5.4 Limit of linear maps 141 142 146 147 147 148 149 151 152 156 157 160 160 161 Linear functionals ... main theorems of linear functional analysis and, at the same time, to provide complete proofs So there is a separate appendix where the Lebesgue integral is introduced in a complete functional analytic