Banach Lattices and Positive Operators
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H erausgegeben von S. S. Chern 1. L. Doob LDouglas, jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf W Maak S. Mac Lane W Magnus M. M. Postnikov F. K. Schmidt D. S. Scott K. Stein
Geschaftsfuhrende H erausgeber B. Eckmann
1. K. Moser
B. L. van der Waerden
Helmut H. Schaefer
Banach Lattices and Positive Operators
Springer-Verlag Berlin Heidelberg New York 1974
Dr. rer. nat. Helmut H. Schaefer Professor of Mathematics, University of Tiibingen
AMS Subject Classification (1970): 06A65, 15A48, 15A51, 46-01, 46A35, 46A40, 46B99, 46E05, 47-01, 47B15, 47B55, 47B99, 47D05, 47D10, 47D15
ISBN·13: 978·3·642·65972·0
c·ISBN·13: 978·3·642·65970·6
001: 10.1007/978·3·642·65970·6 Library of Congress Cataloging in Publication Data Schaefer. Helmut H. Banach lattices and positive operators. (Die Grundlehren def mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtigung def Anwendungsgebiete; Sd. 215)
Bibliography: p. Includes indexes. 1. Linear topological spaces. 2. Lattice theory. 3. Linear operators. I. Title. II. Series: Die Grundlehren def mathematischen Wissenschaften in Einzeldarstellungen: Bd. 215. QA322.S27 515'.73 74-22499 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned. specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer· Verlag Berlin· Heidelberg 1974. Typesetting and printing: Zechnersche Buchdruckerei, Speyer, Bookbinding: Konrad Triltsch, Wiirzburg. Softcover reprint of the hardcover 1st edition 1974
To Rhea, Christoph, and Mark
Preface
Vector lattices-also called Riesz spaces, K-lineals, or linear lattices-were first considered by F. Riesz, L. Kantorovic, and H. Freudenthal in the middle nineteen thirties; thus their early theory dates back almost as far as the beginning of the systematic investigation of Banach spaces. Schools of research on vector lattices were subsequently founded in the Soviet Union (Kantorovic, Judin, Pinsker, Vulikh) and in Japan (Nakano, Ogasawara, Yosida); other important contributions came from the United States (G. Birkhoff, Kakutani, M. H. Stone). L. Kantorovic and his school first recognized the importance of studying vector lattices in connection with Banach's theory of normed vector spaces; they investigated normed vector lattices as well as order-related linear operators between such vector lattices. (Cf. Kantorovic-Vulikh-Pinsker [1950] and Vulikh [1967].) However, in the years following that early period, functional analysis and vector lattice theory began drifting more and more apart; it is my impression that "linear order theory" could not quite keep pace with the rapid development of general functional analysis and thus developed into a theory largel
