Ideal Spaces
Ideal spaces are a very general class of normed spaces of measurable functions, which includes e.g. Lebesgue and Orlicz spaces. Their most important application is in functional analysis in the theory of (usual and partial) integral and integro-differenti
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1664
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1664
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Martin Vath
Ideal Spaces
Springer
Author Martin Vath Mathematisches Institut Universitat Wurzburg Am Hubland D-97074 Wurzburg, Germany e-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Vath, Martin:
Ideal spaces / Martin Vath. - Berlin; Heidelberg; New York; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara; Singapore; Tokyo: Springer, 1997 (Lecture notes in mathematics; 1664) ISBN 3-540-63160-7
Mathematics Subject Classification (1991): Primary: 46E30, 46E40; Secondary: 46A45, 46B45, 28A20, 28A35, 28E15, 46BI0 ISSN 0075-8434 ISBN 3-540-63160-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10553283 46/3142-543210 - Printed on acid-free paper
Table of Contents
1.
Introduction..............................................
1
2.
Basic Definitions and Properties. . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Ideal Spaces and their Characterization. . . . . . . . . . . . . . . . . . . 7 2.2 Extended Convergence and the Support of FUnctions. . . . . . .. 17
3.
Ideal Spaces with Additional Properties . . . . . . . . . . . . . . . . .. 3.1 The W-Property 3.2 Completeness and Perfectness. . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3 Regular Spaces and Convergence Theorems. . . . . . . . . . . . . . .. 3.4 Associate Spaces 3.5 Dual Spaces and Reflexivity
4.
Ideal Spaces on Product Measures and Calculus. . . . . . . . .. 75 4.1 Spaces with Mixed Norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75 4.2 Weighted Spaces and Projections of Spaces . . . . . . . . . . . . . . .. 82 4.3 Spaces with Mixed Family Norm. . . . . . . . . . . . . . . . . . . . . . . .. 85 4.4 Calculus with Ideal-valued Functions. . . . . . . . . . . . . . . . . . . . .. 98
5.
Operators and Applications 105 5.1 Automatic Continuit
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