The Isometric Theory of Classical Banach Spaces
The purpose of this book is to present the main structure theorems in the isometric theory of classical Banach spaces. Elements of general topology, measure theory, and Banach spaces are assumed to be familiar to the reader. A classical Banach space is a
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H erausgegeben von
S. S. Chern J. L. Doob J. Douglas, 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 Geschiiftsfuhrende H erausgeber
B. Eckmann
J. K. Moser
B. L. van der Waerden
H. Elton Lacey
The Isometric Theory of Classical Banach Spaces
Springer-Verlag Berlin Heidelberg New York 1974
H. Elton Lacey University of Texas at Austin, Department of Mathematics Austin, TX 78712/USA
Geschiiftsfiihrende Herausgeber
B. Eckmann Eidgenossische Technische Hochschule Ziirich
J. K. Moser Courant Institute of Mathematical Sciences New York
B. L. van der Waerden
Mathematisches Institut der Universitiit Ziirich
AMS Subject Classification (1970) Primary 46E05, 46E15, 46E25, 46E30 Secondary 46B05, 46J10, 47B05, 47B55
ISBN -13:978-3-642-65764-1 e- ISBN -13:978-3-642-65762-7 DOl: 10.1007/978-3-642-65762-7 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 mWlhine 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. Library of Congress Catalog Card Number 74-394. Softcover reprint of the hardcover 1st edition 1974
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Preface
The purpose of this book is to present the main structure theorems in the isometric theory of classical Banach spaces. Elements of general topology, measure theory, and Banach spaces are assumed to be familiar to the reader. A classical Banach space is a Banach space X whose dual space is linearly isometric to Lp(j1, IR) (or Lp(j1, CC) in the complex case) for some measure j1 and some 1 ~ p ~ 00. If 1 < p < 00, then it is well known that X=L q(j1,IR) where 1/p+1/q=1 and if p=oo, then X=L j (v,lR) for some measure v. Thus, the only case where a space is obtained which is not truly classical is when p = 1. This class of spaces is known as L 1 predual spaces since their duals are L j type. It includes some well known subclasses such as spaces of the type C(T, IR) for T a compact Hausdorff space and abstract M spaces. The structure theorems concern necessary and sufficient conditions that a general Banach space is linearly isometric to a classical Banach space. They are framed in terms of conditions on the norm of the space X, conditions on the dual space X*, and on (finite dimensional) subspaces of X. Since most of these spaces are Banach lattices and Banach algebras, characterizations among theses classes are also given. Both the real and complex cases are treated in general (however, some of the geometric results are for the real case only). There is, of course, a corresponding isomorphic theory of classical Banach spaces which is under intensive investigation. This theory studies properties of t
