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M. Scott Osborne Locally Convex Spaces Graduate Texts in Mathematics 269 Graduate Texts in Mathematics Graduate Texts in Mathematics M. Scott Osborne Locally Convex Spaces Graduate Texts in Mathematics 269 Graduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA Advisory Board: Colin Adams, Williams College, Williamstown, MA, USA Alejandro Adem, University of British Columbia, Vancouver, BC, Canada Ruth Charney, Brandeis University, Waltham, MA, USA Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA Roger E. Howe, Yale University, New Haven, CT, USA David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA Jill Pipher, Brown University, Providence, RI, USA Fadil Santosa, University of Minnesota, Minneapolis, MN, USA Amie Wilkinson, University of Chicago, Chicago, IL, USA Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study. For further volumes: http://www.springer.com/series/136 M. Scott Osborne Locally Convex Spaces 123 M. Scott Osborne Department of Mathematics University of Washington Seattle, WA, USA ISSN 0072-5285 ISSN 2197-5612 (electronic) ISBN 978-3-319-02044-0 ISBN 978-3-319-02045-7 (eBook) DOI 10.1007/978-3-319-02045-7 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013951652 Mathematics Subject Classification: 46A03, 46-01, 46A04, 46A08, 46A30 © Springer International Publishing Switzerland 2014 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Functional analysis is an exceptionally useful subject, which is why a certain amount of it is included in most beginning graduate courses on real analysis. For most practicing analysts, however, the restriction to Banach spaces is not enough. This book is intended to cover most of the general theory needed for application to other areas of analysis. Most books on functional analysis come in one of two types: Either they restrict attention to Banach spaces or they cover the general theory in great detail. For the kind of courses I have taught, those of the first type don’t cover a broad enough range of spaces, while those of the second type cover too much material for the time allowed. Don’t misunderstand; there is plenty of interesting material in both kinds, and the interested party is invited to check them out. In fact, in this book, some side topics (e.g., topological groups or Mahowald’s theorem) are treated, provided they don’t go too far afield from the central topic. Also, some useful things are included that are hard to find elsewhere: Sect. 5.2 contains results that are not new but are not covered in most treatments (although Yosida [41] comes very close), for example. The material here is based on a course I taught at the University of Washington. The course lasted one quarter and covered most of the material through Chap. 5. I was using Rudin [32] as a text. His early coverage of locally convex spaces is excellent, but he discusses dual spaces only for Banach spaces. I used his book to organize things in the course but had to expand considerably on the subject matter. (Rudin’s book, by the way, is one of the few that does not fit the “two types” dichotomy above. Neither do Conway [7] or Reed and Simon [29].) The prerequisite for this book is the Banach space theory typically taught in a beginning graduate real analysis course. The material in Folland [15], Royden [30], or Rudin [31] cover it all; Bruckner, Bruckner, and Thomson [6] cover everything except the Riesz representation theorem for general compact Hausdorff spaces, and that appears here only in an application in Sect. 5.5 and in Appendix C.Thereare some topological results that are needed, which may or may not have appeared in a beginning graduate real analysis course; these appear in Appendix A. One more thing. Although this book is oriented toward applications, the beauty of the subject may appeal to you. If so, there is plenty out there to v vi Preface look at: Edwards [13], Grothendieck [16], Horvath [18], Kelley and Namioka [21], Schaefer [33], and Treves [36] are good reading on the general subject. Bachman and Narici [2], Narici and Beckenstein [27], and Schechter [34] do a nice job with Banach spaces. Wilansky [39] updates with more modern concepts the functional analysts have come up with (e.g., “webbed spaces”). Edwards [13], Phelps [28], Reed and Simon [29], and Treves [36] have plenty of material on the interaction of functional analysis with outside subjects. Also, Swartz [35] and Wong [40]are recommended. Finally, Dieudonne [10] gives a very readable history of the subject. Finally, some “thank you’s”: To Prof. Garth Warner and Clayton Barnes, for comments and suggestions. To Owen Biesel, Nathaniel Blair-Stahn, Ryan Card, Michael Gaul, and Dustin Mayeda, who took the original course. And (big time!) to Mary Sheetz, who put the manuscript together. And a closing thank you to Elizabeth Davis, Richard Kozarek, Ronald Mason, Michael Mullins, Huong Pham, Vincent Picozzi, Betsy Ross, Chelsey Stevens, Megan Stewart, and L. William Traverso, without whose aid this manuscript would never have been completed. Seattle, WA, USA M. Scott Osborne Contents 1 Topological Groups .......................................................... 1 1.1 Point Set Topology ...................................................... 1 1.2 Topological Groups: Neighborhood Bases ............................ 7 1.3 Set Products ............................................................. 13 1.4 Constructions ............................................................ 15 1.5 Completeness............................................................ 23 1.6 Completeness and First Countability................................... 25 2 Topological Vector Spaces................................................... 33 2.1 Generalities .............................................................. 33 2.2 Special Subsets .......................................................... 37 2.3 Convexity ................................................................ 41 2.4 Linear Transformations ................................................. 45 3 Locally Convex Spaces ...................................................... 51 3.1 Bases..................................................................... 51 3.2 Gauges and Seminorms ................................................. 53 3.3 The Hahn–Banach Theorem ............................................ 57 3.4 The Dual ................................................................. 61 3.5 Polars .................................................................... 63 3.6 Associated Topologies .................................................. 66 3.7 Seminorms and Fréchet Spaces ........................................ 74 3.8 LF-Spaces ............................................................... 80 4 The Classics................................................................... 95 4.1 Three Special Properties ................................................ 95 4.2 Uniform Boundedness .................................................. 100 4.3 Completeness............................................................ 105 4.4 The Open Mapping Theorem ........................................... 110 4.5 The Closed Graph Theorem ............................................ 114 vii viii Contents 5 Dual Spaces ..................................................................
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