Functional Analysis
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Herausgegeben von
J. L. Doob · E. Heinz · F. Hitzebruch · E. Hopf · H. Hopf W. Maak · S. MacLane · W. Magnus · D. Mumford M. M. Postnikov · F. K. Schmidt · D. S. Scott · K. Stein
Geschäftsführende Herausgeber B. Eckmann und B. L. van der Waerden
K8saku Yosida
Functional Analysis Second Edition
Springer-Verlag Berlin Heidelberg GmbH 1968
Prof. KOsaku Yosida Department of Mathematics, University of Tokyo
Geschliftsfilhrende Herausgeber:
Prof. Dr. B. Eckmann EidgenlSssische Technische Hochscbule Zilrich
Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitlit Zilrich
ISBN 978-3-662-11791-0 (eBook) ISBN 978-3-662-11793-4 DOI 10.1007/978-3-662-11791-0
Alle Rechte vorbehalten. Kein Teil dieses Buches darf ohne schriftliche Genehmigung des Springer-Verlages iibersetzt oder in irgendeiner Form vervielfăltigt werden
© by Springer-Verlag Berlin Heidelberg
1965 and 1968
Originally published by Springer-Verlag, Berlin Heidelberg in 1968
Softcover reprint of the hardcover 1st edition 1968 Library of Congress Catalog Card Number 64-8025
Titei-Nr. 5106
Preface to the First Edition The present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook tobe studied by students on their own or tobe used in a course on Functional Analysis, i.e., the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis. Necessary prerequisites for the reading of this book are summarized, with or without proof, in Chapter 0 under titles: Set Theory, Topological Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Rilbert spaces is presented in connection with the theory of generalized functions of S. L. SoBOLEV and L. ScHwARTZ. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathematicians, both pure and applied. The reader may pass e.g. from Chapter IX (Analyti{:al Theory of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration of the Equation of Evolution). Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X, respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators. In the preparation of the present· book, the author has received valuable advice and criticism from many friends. Especially, Mrs. K. RILLE has kindly read through the manuscript as well as the galley and page proofs. Without her painstaking help, this book could not have been printed in the present style in the language which was not spoken to the author in the cradle. The author owes very much to his old friends, Professor E. RILLE and Professor S. KAKUTANI of Yale University and Professor R. S. PHILLIPS of Stanford Univer
