Twisted Sums of Banach and Nuclear Spaces

Abstract

A twisted sum of (topological vector) spaces Y and Z is a space X with a subspace Y1 isomorphic to Y for which X/Y1 is isomorphic to Z. It splits if Y, is complemented. It is proved that every twisted sum of a Banach space Y and a nuclear space Z splits. Kothe sequence spaces Z for which this holds are characterized. Every locally convex twisted sum of a nuclear Frechet space Y and a Banach space Z splits too. If Z is superreflexive, then the local convexity assumption on the twisted sum may be omitted. Other results of this kind on K6the sequence spaces are obtained. Introduction. In this paper topological vector spaces (tvs) and locally convex spaces (lcs) are generally not assumed to be Hausdorff. We define precisely twisted sums as follows: a diagram of tvs and continuous, relatively open linear maps,