II.E - Local properties

Summary

In this chapter we present some results and notions concerning finite dimensional Banach spaces and the relation between an infinite dimensional Banach space and its finite dimensional subspaces. We start with a discussion of the bounded approximation property and the TTA-spaces. We also prove the local reflexivity principle which connects the local properties of X and X**. We prove the Auerbach lemma which allows a good identification of an n-dimensional Banach space with ℝⁿ or ℂⁿ. We also study the concept of Banach-Mazur distance.

1. By local properties of a Banach space we mean the properties which depend on the structure of finite dimensional subspaces of the space. Some examples of such properties will be pointed out in this chapter and many more will be encountered in the sequel.

The basic aim of this chapter is to provide an elementary understanding of local phenomena. Even at this early stage it is apparent that one needs a clarification of two points:

1. (a) how the general Banach space is built up from finite dimensional subspaces;

2. (b) what are the relevant properties of finite dimensional spaces.

Let us start with some definitions and examples which explain point (a) a little. What we are really thinking about in (a) is the approximation problem: how well can we approximate the identity operator on the space X by finite dimensional operators?