Decomposition of Banach Space into a Direct Sum of Separable and Reflexive Subspaces and Borel Maps

Abstract

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 be decomposed into a direct sum of separable and reflexive subspaces, then there exists a normed space Z and a linear continuous bijective operator T : E → Z such that T^(−1) is not a Borel map.