The Wijsman Convergence

Abstract

In 1960's, when he studied some optimum properties of sequential probability ratio test, R. A. Wijsman considered a mode of convergence for sequences of closed sets such that if a sequence of proper lower semicontinuous convex functions defined on R^n (as associated with their epigraphs) converged, then the same could be said for the induced sequence of conjugate functions. In recognition of his contribution, this mode of convergence is now called the Wijsman convergence. Of course, the Wijsman convergence is topological, and the associated topology is called the Wijsman topology. In the past 40 years, there has been a considerable effort in extending Wijsman's results in infinite-dimensional Banach spaces, and exploring some properties of the Wijsman topology. It turns out that the Wijsman convergence and its topology have played important roles in the Banach Space Theory and Set-Valued Analysis. In this talk, I shall give an overview of the recent developments on the Wijsman topology. In particular, I shall discuss the Polishness, Baireness, the Amsterdam and other properties of this topology. Parts of my recent work with H Junnila and A Tomita will be presented, and some open questions in this area will be posed.