Operator-Valued Measures and Integrals for Cone-Valued Functions
Integration theory deals with extended real-valued, vector-valued, or operator-valued measures and functions. Different approaches are applied in each of these cases using different techniques. The order structure of the (extended) real number system
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1964
Walter Roth
Operator-Valued Measures and Integrals for Cone-Valued Functions
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Walter Roth Department of Mathematics University of Brunei Darussalam BE 1410 Gadong Brunei Darussalam [email protected]
ISBN: 978-3-540-87564-2 e-ISBN: 978-3-540-87565-9 DOI: 10.1007/978-3-540-87565-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008938191 Mathematics Subject Classification (2000): 28B20, 46A13, 46E40, 46G10 c 2009 Springer-Verlag Berlin Heidelberg 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. 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. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com
Preface
The aim of this book is twofold: Firstly, to introduce the developing theory of locally convex cones to a wider audience. This theory generalizes locally convex topological vector spaces and permits many additional and substantially different examples and applications. In the aspects of the theory that have been developed so far, the increase in generality does not lead to any compromises with respect to the depth of its results. The main difference to vector spaces is the presence of infinity-type unbounded elements and the general non-availability of the cancellation law. Some important mathematical models, while close to the structure of vector spaces are of this type. They do not allow subtraction of their elements or multiplication by negative scalars. Examples are certain classes of set-valued or extended real-valued functions that may take infinite values. These arise naturally in integration theory, potential theory and in a variety of other settings and do not form vector spaces. Therefore many results and techniques from classical functional analysis can not be immediately applied. Locally convex cones carry a reflexive and transitive order relation, and their (convex semiuniform) topology is defined using this order structure. The first part of this book contains a review and summary of the aspects of the theory of locally convex cones that have been developed so far, sometimes without detailed proofs, but references to the sources instead. The theory is then developed further, adding some (hopefully) interesting new features. This leads
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