* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95.The main results of the paper are: Theorem 1. Let a Banach space E be decomposed into a direct sum of separable and reflexive subspaces. Then for every Hausdorff locally convex topological vector space Z and for every linear continuous bijective operator T : E → Z, the inverse T^(−1) is a Borel map. Theorem 2. Let us assume the continuum hypothesis. If a Banach space E cannot...

We study the class of descriptive compact spaces, the Banach spaces generated by descriptive compact subsets and their relation to renorming problems.

If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d̂({\overline{co}}^{w*}\left(K\right),C):=supinf\left|\right|k-c\left|\right|:c\in C:k\in {\overline{co}}^{w*}\left(K\right)$ from ${\overline{co}}^{w*}\left(K\right)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d̂({\overline{co}}^{w*}\left(K\right),C)\le Md̂(K,C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into...