WEAK AND STRONG CONVERGENCE IN THE HYPERSPACE CC(X)

Abstract

Let $CC(X)$ be the collection of all non-empty compact, convex subsets of a complex Banach space $X$ endowed with the usual Hausdorff metric $h$. We shall define a natural weak topology ${\cal T}_w$ on $CC(X)$ and investigate properties of ${\cal T}_w$-convergent sequences. Our main result is a theorem which states that if $A_n$, $A\in CC(X)$ and $A_n$ is ${\cal T}_w$-convergent to $A$, then there exists a sequence $\{B_n\}$ (each $B_n$ is a finite convex combination of $A_k$'s) such that $B_n$ converges to $A$ with respact to the Hausdorff metric $h$.