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This text is based on lectures given by the author at the advanced undergaraduate and graduate levels in Measure theory, Functional Analysis, Bannach Algebras, Spectral Theory (of bounded and unbounded operators), Semigroups of Operators, Probability and Mathematical Statistics, and Partial Differential Equations. The first 10 chapters discuss theoretical methods in Measure Theory and Functional Analysis and contain over 120 end of chapter exercises. the final two chapters apply theory to applications in Probability Theory and Partial Differential Equations.
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Oxford Graduate Texts in Mathematics Series Editors R. Cohen S. K. Donaldson S. Hildebrandt T. J. Lyons M. J. Taylor oxford graduate texts in mathematics 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Keith Hannabuss: An Introduction to Quantum Theory Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis James G. Oxley: Matroid Theory N. J. Hitchin, G. B. Segal, and R. S. Ward: Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces Wulf Rossmann: Lie Groups: An Introduction Through Linear Groups Q. Liu: Algebraic Geometry and Arithmetic Curves Martin R. Bridson and Simon M, Salamon (eds): Invitations to Geometry and Topology Shmuel Kantorovitz: Introduction to Modern Analysis Terry Lawson: Topology: A Geometric Approach Meinolf Geck: An Introduction to Algebraic Geometry and Algebraic Groups Introduction to Modern Analysis Shmuel Kantorovitz Bar Ilan University, Ramat Gan, Israel 1 3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Oxford University Press 2003 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2003 First published in paperback 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn ISBN 0–19–852656–3 ISBN 0–19–920315–6 (Pbk.) 978–0–19–852656–8 978–0–19–920315–4 (Pbk.) 1 3 5 7 9 10 8 6 4 2 To Ita, Bracha, Pnina, Pinchas, and Ruth This page intentionally left blank Preface This book grew out of lectures given since 1964 at Yale University, the University of Illinois at Chicago, and Bar Ilan University. The material covers the usual topics of Measure Theory and Functional Analysis, with applications to Probability Theory and to the theory of linear partial differential equations. Some relatively advanced t