5 - Local Methods in the Theory of Twisted Sums

Summary

Just as there is a local theory of Banach spaces, there is a local theory of exact sequences of (quasi-) Banach spaces. In this chapter we explain what it means and how it can be used. Following the usage of Banach space theory, `local’ refers to finite-dimensional objects, and so we consider exact sequences that split locally; i.e. they split at the finite-dimensional level. The material of the chapter is divided into three sections. The first contains the definition and characterisations of locally split sequences and their connections with the extension and lifting of operators. The second presents the uniform boundedness theorem for exact sequences. The third is devoted to applications: under quite natural hypotheses, it is shown that $\operatorname{Ext}(X, Y)=0$ implies that also $\operatorname{Ext}(X’, Y’)=0$ when $X’$ has the same local structure as $X$ and $Y’$ has the same local structure as $Y$. From here we can easily obtain that $\operatorname{Ext}(X, Y)\neq 0$ for many pairs of spaces $X,Y$, both classical and exotic.