Lindelöf property in function spaces and a related selection theorem

Abstract:

Let $X$ be a separable metrizable space. If $K$ is a compact space whose function space $C(K)$ is weakly $\mathcal {K}$-analytic, then the space ${C_p}(X,K)$ of continuous maps from $X$ to $K$ with the pointwise topology has the Lindelöf property. If $E$ is a Banach space whose weak topology is $\mathcal {K}$-analytic, then each lower semicontinuous map from $X$ to the family of nonempty closed convex subsets of the unit ball of the dual $E$ with the weak*-topology admits a continuous selection. This extends some results of Corson and Lindenstrauss.