From Classical to Modern Analysis
This innovative textbook bridges the gap between undergraduate analysis and graduate measure theory by guiding students from the classical foundations of analysis to more modern topics like metric spaces and Lebesgue integration. Designed for a two-semest
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From Classical to Modern Analysis
Rinaldo B. Schinazi
From Classical to Modern Analysis
Rinaldo B. Schinazi Department of Mathematics University of Colorado Colorado Springs, CO, USA
ISBN 978-3-319-94582-8 ISBN 978-3-319-94583-5 (eBook) https://doi.org/10.1007/978-3-319-94583-5 Library of Congress Control Number: 2018950239 Mathematics Subject Classification (2010): 26-01, 28-01, 40A05 © Springer International Publishing AG, part of Springer Nature 2018 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. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
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Preface
Why Another Analysis Book? There are many good undergraduate and graduate analysis books. There is, however, a rather large gap between undergraduate and graduate texts. The purpose of this book is to bridge this gap. Typically, a graduate book covers measure theory and the Lebesgue integral. An undergraduate book does not. A book covering measure theory usually assumes that the reader is familiar with limits superior and inferior, arithmetic operations involving infinity, set theory, metric spaces, and some topology. Many students get into a measure theory course with only a vague notion of these concepts. Even more importantly, the level of abstraction the students have been exposed to is much lower than what is required to succeed in a graduate analysis course. As a result measure theory courses are much harder for the student than they need to be. The main goal of this book is to propose a gently rising path from undergraduate analysis (i.e., Classical Analysis) to measure theory (i.e., Modern Analysis). Prerequisites We assume the reader of this text to have had an advanced calculus course a
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