Abstract

Let B(X,Y)\mathbf{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)\mathbf{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×YX\times Y.

Author information

Contact details are reproduced from the original publication and may be historical.

Gerald Beer

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

gbeer@cslanet.calstatela.edu

Keywords

  • Operator topology
  • polar topology
  • bornological convergence
  • Attouch-Wets Convergence
  • normed linear space
  • convex set
  • starshaped set

Mathematics Subject Classification

47A05; 46A17, 54B20

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Suggested citation

G. Beer. “Operator Topologies and Graph Convergence.” Journal of Convex Analysis 16 (2009), No. 3&4, 687–698. https://doi.org/10.68381/jca16040

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press.