CONVERGENCES FOR SEQUENCES OF SETS AND LINEAR MAPPINGS

Let E, F be normed vector spaces and let A, A, be continuous linear op erators from E into F. Consider X, X(n) [Y, Y-n] arbitrary nonempty cl osed subsets of E [F]. We study the problem: if X = lim X(n) [Y = lim Y-n] in the sense of Kuratowski (or Mosco, or Attouch-Wets), do we hav e A(X) = lim A(X,) [A(-1)(Y) = lim A(n)(-1)(Y-n)] in the same sense ? In solving this problem we improve recent results obtained by I. Schoc hetman and R. Smith and published in this journal, and extend some res ults of G. Beer established for the case F = R. (C) 1994 Academic Pres s, Inc.