Convergences for Sequences of Sets and Linear Mappings

Let E, F be normed vector spaces and let A, An be continuous linear operators from E into F. Consider X, Xn [Y, Yn] arbitrary nonempty closed subsets of E [F]. We study the problem: if X = lim Xn [Y = lim Yn] in the sense of Kuratowski (or Mosco, or Attouch-Wets), do we have A(X) = lim A(Xn) [A−1(Y) = lim A−1n(Yn)] in the same sense ? In solving this problem we improve recent results obtained by I. Schochetman and R. Smith and published in this journal, and extend some results of G. Beer established for the case F = R.