Dual Spaces, Transposes and Adjoints
In this chapter we develop a duality between a normed space X and the space \(X'\) consisting of all bounded linear functionals on X, known as the dual space of X. As a consequence of the Hahn–Banach extension theorem, we show that \(X'\ne \{0\}\) if \(X\
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Linear Functional Analysis for Scientists and Engineers
Linear Functional Analysis for Scientists and Engineers
Balmohan V. Limaye
Linear Functional Analysis for Scientists and Engineers
123
Balmohan V. Limaye Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai India
ISBN 978-981-10-0970-9 DOI 10.1007/978-981-10-0972-3
ISBN 978-981-10-0972-3
(eBook)
Library of Congress Control Number: 2016939047 © Springer Science+Business Media Singapore 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 Science+Business Media Singapore Pte Ltd.
Preface
The aim of this book is to provide a short and simple introduction to the charming subject of linear functional analysis. The adjective ‘linear’ is used to indicate our focus on linear maps between linear spaces. The applicability of the topics covered to problems in science and engineering is kept in mind. In principle, this book is accessible to anyone who has completed a course in linear algebra and a course in real analysis. Relevant topics from these subjects are collated in Chap. 1 for a ready reference. A familiarity with measure theory is not required for the development of the main results in this book, but it would help appreciate them better. For this reason, a sketch of rudimentary results about the Lebesgue measure on the real line is included in Chap. 1. To keep the prerequisites minimal, we have restricted to metric topology, and to the Lebesgue measure, that is, neither general topological spaces nor arbitrary measure spaces would be considered. Year after year, several students from the engineering branches, who took my course on functional analysis at the Indian Institute of Technology Bombay, have exclaimed ‘Why were we not introduced to this subject in the early stages of our work? Many things with which we had to struggle hard would have fallen in place right at the sta
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