The Convenient Setting of Global Analysis

Abstract

The aim of this book is to lay foundations of differential calculus in infinite dimensions and to discuss those applications in infinite dimensional differential geometry and global analysis which do not involve Sobolev completions and fixed point theory. The approach is very simple: A mapping is called smooth if it maps smooth curves to smooth curves. All other properties are proved results and not assumptions: Like chain rule, existence and linearity of derivatives, powerful smooth uniformly boundedness theorems are available. Up to Fréchet spaces this notion of smoothness coincides with all known reasonable concepts. In the same spirit calculus of holomorphic mappings (including Hartogs’ theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups. Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations. Corrections and complements to this book will be posted on the internet at the URL http://www.mat.univie.ac.at/~michor/apbook.ps Library of Congress Cataloging-in-Publication Data Kriegl, Andreas. The convenient setting of global analysis / Andreas Kriegl, Peter W. Michor. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 53) Includes bibliographical references (p. – ) and index. ISBN 0-8218-0780-3 (alk. paper) 1. Global analysis (Mathematics) I. Michor, Peter W., 1949– . II. Title. III. Series: Mathematical surveys and monographs ; no. 53. QA614.K75 1997 514′.74—dc21 97-25857 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams.org. c © 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ©∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS homepage at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 02 01 00 99 98 97

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@inproceedings{Michor1997TheCS, title={The Convenient Setting of Global Analysis}, author={Peter W. Michor and Andreas Kriegl}, year={1997} }