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Function spaces and the Mosco topology

Published online by Cambridge University Press:  17 April 2009

Gerald Beer  and
Robert Tamaki
Gerald Beer
Affiliation:
Department of Mathematics, California State University, Los Angeles Los Angeles, CA 90032, United States of America
Robert Tamaki
Affiliation:
Department of Mathematics, California State University, Los Angeles Los Angeles, CA 90032, United States of America
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Abstract

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Let X and Y be Banach spaces and let C(X, Y) be the functions from X to Y continuous with respect to the weak topology on X and the strong topology on Y. By the Mosco topology τM on C(X, Y) we mean the supremum of the Fell topologies determined by the weak and strong topologies on X × Y, where functions are identified with their graphs. The function space is Hausdorff if and only if both X and Y are reflexive. Moreover, τM coincides with the stronger compact-open topology on C(X, Y) provided X is reflexive and Y is finite dimensional. We also show convergence in either sense is properly weaker than continuous convergence, even for continuous linear functionals, whenever X is infinite dimensional. For real-valued weakly continuous functions, τM is the supremum of the Mosco epitopology and the Mosco hypotopology if and only if X is reflexive.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 1 , August 1990 , pp. 7 - 19
Copyright
Copyright © Australian Mathematical Society 1990

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