Extenders for vector-valued functions

Given a subset A of a topological space X, a locally convex space Y, and a family C of subsets of Y we study the problem of the existence of a linear C-extender u : C infinity(A, Y) -> C infinity(X, Y), which is a linear operator extending bounded continuous functions f : A -> C subset of Y, C is an element of C, to bounded continuous functions (f) over bar = U(f) : X -> C subset of Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game The results obtained allow us to characterize reflexive Banach spaces as the only normed spaces Y such that for every closed subset A of a GO-space X there is a C-extender U : C infinity(A, Y) -> C infinity(X, Y) for the family C of closed convex subsets of Y. Also we obtain a characterization of Polish spaces and of weakly sequentially complete Banach lattices in terms of extenders.