Extenders for vector-valued functions
Given a subset $A$ of a topological space $X$, a locally convex space $Y$, and a family $\mathbb C$ of subsets of $Y$ we study the problem of the existence of a linear $\mathbb C$-extender $u:C_\infty(A,Y)\to C_\infty(X,Y)$, which is a linear operator extending bounded continuous functions $f:A\to C\subset Y$, $C\in\mathbb C$, to bounded continuous functions $\overline f =u(f):X\to C\subset Y$. 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 $\mathbb C$-extender $u:C_\infty(A,Y)\to C_\infty(X,Y)$ for the family $\mathbb C$ 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.