Orlicz’s Theorem and The Structure of Finite-Dimensional Subspaces

Abstract

In the preceding chapters we have often encountered assertions of the following type: from a certain property of the sum of a finite number of elements follows a certain theorem on series in an infinite-dimensional space. Since any finite set of elements lies in some finite-dimensional subspace, in all such assertions only special features of the structure of the finite-dimensional subspaces of the ambient infinite-dimensional space considered play a role. In the first section of this chapter we will present a modern convenient language for describing what kind of finite-dimensional subspaces a given infinite-dimensional space has. In the sequel we shall actively employ this language and, in particular, give a complete description of the spaces in which an analogue of Orlicz’s theorem on unconditionally convergent series holds.