3 - Quasilinear Maps

Summary

In this chapter we plunge into the non-linear aspects of the theory of twisted sums. One of the objectives of this chapter is to provide the reader with practical ways to construct non-trivial exact sequences $0 \longrightarrow Y \longrightarrow \cdot \longrightarrow X \longrightarrow 0$ when only the spaces $Y$ and $X$ are known. The central idea here is that such exact sequences correspond to a certain type of non-linear map called a quasilinear map $\Phi: X \longrightarrow Y$. The chapter has been organised so that the reader can reach at an early stage a number of important applications. The topics covered include finding pairs of quasi-Banach spaces $X, Y$ such that all exact sequences $0 \longrightarrow Y \longrightarrow \cdot \longrightarrow X \longrightarrow 0$ split, natural representations for the functor $\operatorname{Ext}$, getting valuable insight into the structure of exact sequences and twisted sum spaces, a duality theory for exact sequences of Banach spaces (including a non-linear Hahn-Banach theorem), uniform boundedness principles for exact sequences leading to a local theory for exact sequences, homological properties of the spaces $\ell_p$ and $L_p$, type of twisted sums, $\mathscr K$-spaces and the Kalton-Peck maps.