Metric Modular Spaces
Aimed toward researchers and graduate students familiar with elements of functional analysis, linear algebra, and general topology; this book contains a general study of modulars, modular spaces, and metric modular spaces. Modulars may be thought of as ge
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Vyacheslav V. Chistyakov
Metric Modular Spaces Theory and Applications
123
SpringerBriefs in Mathematics
Series Editors Nicola Bellomo Michele Benzi Palle E. T. Jorgensen Tatsien Li Roderick Melnik Otmar Scherzer Benjamin Steinberg Lothar Reichel Yuri Tschinkel G. George Yin Ping Zhang
SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. More information about this series at http://www.springer.com/series/10030
Vyacheslav V. Chistyakov
Metric Modular Spaces Theory and Applications
123
Vyacheslav V. Chistyakov National Research University Higher School of Economics Department of Informatics, Mathematics and Computer Science Nizhny Novgorod, Russia
ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-319-25281-0 ISBN 978-3-319-25283-4 (eBook) DOI 10.1007/978-3-319-25283-4 Library of Congress Control Number: 2015956774 Mathematics Subject Classification (2010): 46A80, 54E35, 26A45, 47H30, 26E25, 34A12 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 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 Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
To my family with love: Zinaida and Vasili˘ı (parents) Svetlana (wife) Dar’ya and Vasilisa (daughters)
Preface
The theory of metric spaces was created by Fréchet [39] and Hausdorff [43] a century ago. In its basis is the notion of distance between any two points of a set. Usually (but not necessarily), the algebraic structure of the set does not play any role in the metric space analysis. If the set under consideration has a ric
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