Topological 0-1 laws for subspaces of a Banach space with a Schauder basis

Abstract

For a Banach space $X$ with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of $X$ as well as for subspaces of $X$ with a successive finite-dimensional decomposition on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional basis which has a complemented subspace without an unconditional basis, are deduced.