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Journal of Convex Analysis 16 (2009), No. 3, 687--698
Copyright Heldermann Verlag 2009

Operator Topologies and Graph Convergence

Gerald Beer
Department of Mathematics, California State University, 5151 State University Drive, Los Angeles, CA 90032, U.S.A.

Let B(X,Y) be the continuous linear transformations from a normed linear space X to a normed linear space Y. This article presents two general results -- one for the norm topology on Y and one for the weak topology on Y -- that explain how convergence of sequences in B(X,Y) with respect to a topology of uniform convergence on a prescribed family of norm bounded subsets of X is reflected in the bornological convergence of the associated sequence of graphs with respect to a family of subsets of the Cartesian product X times Y.

Keywords: Operator topology, polar topology, bornological convergence, Attouch-Wets Convergence, normed linear space, convex set, starshaped set.

MSC: 47A05; 46A17, 54B20

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