Boundedly complete basic sequences, c ₀-subspaces, and injections of Banach spaces

We study the connection between topological properties of subsets of a given Banach space and their images under linear, continuous one-to-one mappings on the one hand and the existence in a given Banach space of either a boundedly complete basic sequence (BCBS) or an isomorphic copy of c _o (c _o -subspace) on the other hand. We present criteria for the existence of a BCBS. They are deduced from new characterisations of G _δ-embeddings which we also present. We obtain a necessary and sufficient condition for separability of a dual Banach space in terms of saturation by BCBS. Criteria for the existence in a Banach space of a c _o -subspace are also presented. We describe the class of separable Banach spaces which contains either a BCBS or a c _o -subspace.