This chapter studies singular sequences, namely exact sequences whose quotient maps are strictly singular operators. Different methods of construction and examples are presented.

Anyone familiar with $\ell_p$ spaces can follow a healthy 50 per cent of this book; if familiar with $L_p$ spaces, the percentage raises to 75 per cent. All the rest can be found in the text. Anyway, a reasonable list of prerequisites that could help a smooth reading would be some acquaintance with classical Banach space theory; lack of fear when local convexity disappears; a certain bias towards abstraction; calm when non-linear objects show off, and some fondness for exotic spaces. The reader is reminded in this chapter about notation for the book, sets and functions, Boolean algebras, ordinals and cardinals, compact spaces, quasinormed spaces and operators, classical spaces, approximation properties and operator ideals.

The chapter is devoted to the single topic of extending $\mathscr C$-valued operators. Its first section presents the global approach to the extension of operators: Zippin’s characterisation of $\mathscr C$-trivial embeddings by means of weak*-continuous selectors and a few noteworthy applications. The second section presents the Lindenstrauss-Pe\l czy\’nski theorem with two different proofs: the first one combines homological techniques with the global approach, while the second is Lindenstrauss-Pe\l czy\’nski’s original proof. The analysis of their proof is indispensable for understanding Kalton’s imaginative inventions that lead to the so-called $L^*$ and $m_1$-type properties and to a decent list of $\mathscr C$-extensible spaces. The next two sections contain, respectively, those points of the Lipschitz theory that are necessary to develop the linear theory and different aspects of Zippin’s problem: which separable Banach spaces $X$ satisfy $\operatorname{Ext}(X, C(K))=0\,$? The problem admits an interesting gradation in terms of the topological complexity of $K$. The final section reports the complete solution of the problem of whether $\operatorname{Ext}(C(K), c_0)\neq 0$ for all non-metrisable compacta $K$.

Fra\“iss\’e sequences and their limits are universal constructions whose impact on functional analysis and Banach space theory is not yet well appreciated. Our rather pedestrian approach is aimed at the construction and study of two concrete examples: the $p$-Gurariy space, namely the only separable $p$-Banach space of almost universal disposition, and the $p$-Kadec space, a separable $p$-Banach space of almost universal complemented disposition with a 1-FDD. The chapter emphasises that these spaces correspond to the same object, but in different categories.

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.

This chapter focuses on the possibility of extending isomorphisms or isometries to maps of the same type. It presents all known results about the automorphic space problem of Lindenstrauss and Rosenthal, including a dichotomy theorem, and about spaces of universal disposition already envisioned by Gurariy. It also treats finite-dimensional variations of those properties: the rich theory of UFO spaces and finitely automorphic quasi-Banach spaces. The topics of how many positions a Banach space can occupy in a bigger superspace and how many twisted sums of two spaces exist are considered.

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.

The chapter contains the fundamental results about Banach and quasi-Banach spaces and their complemented subspaces that are necessary for this book. Classical topics included are the Aoki-Rolewicz theorem, the completion of a quasinormed space, $p$-Banach envelopes, Pe\l czy\’nski’s decomposition method, uncomplemented subspaces of classical spaces, indecomposable spaces, type and cotype of quasi-Banach spaces, local properties, ultraproducts, the Dunford-Pettis and Grothendieck properties, properties (V) of Pe\l czy\’nski and Rosenthal, $C(K)$-spaces and their complemented subspaces and so on. More advanced topics have been also included, such as Sobczyk’s theorem and its non-separable derivatives and ultrapowers, mainly of the $L_p$-spaces.

The chapter introduces the basic elements of the homological language and translates the statements about complemented and uncomplemented subspaces presented in Chapter 1 into this language. The reader will find everything they need to know at this stage about exact sequences, categorical and homological constructions for absolute beginners and injective and projective Banach and quasi-Banach spaces. The chapter describes the basic homological constructions appearing in nature: complex interpolation, the Nakamura-Kakutani, Foia\c{s}-Singer, Pe\l czy\’nski-Lusky and Bourgain’s $\ell_1$ sequences, the Ciesielski-Pol, Bell-Marciszewski and Bourgain-Pisier constructions, the Johnson-Lindenstrauss spaces and so on. A good number of advanced topics are included: diagonal and parallel principles for exact sequences, the Device, 3-space results, extension and lifting of operators, $M$-ideals and vector-valued Sobczyk’s theorems

The final chapter of the book returns to the place the journey started: classical Banach space theory, with a twist. We can now provide solutions, or at least a better understanding, for a number of open problems. Among the topics covered, the reader will encounter vector-valued forms of Sobczyk’s theorem, isomorphically polyhedral $\mathscr L_\infty$-spaces, Lipschitz and uniformly homeomorphic $\mathscr L_\infty$-spaces, properties of kernels of quotient operators from $\mathscr L_1$-spaces, sophisticated 3-space problems, the extension of $\mathscr L_\infty$-valued operators, Kadec spaces, Kalton-Peck spaces and, at last, the space $Z_2$. All these topics can be easily considered as part of classical Banach space theory, even if the techniques we employ involve most of the machinery developed throughout the book.

This chapter lights from a categorical perspective many of the results treated in previous chapters. Contrary to its notorious reputation, category theory helps in understanding concrete constructions, leads to the right questions and, oftentimes, suggests answers. Categories are used in an elementary way but without sacrificing rigour. The topics covered include the functor $\operatorname{Ext}$, the natural equivalence between $\operatorname{Ext}$ and the spaces of quasilinear maps studied in Chapter 3 (including the categorical meaning of `natural’) and the form in which all the pieces fit together in longer exact sequences and their uses, adjoint and derived functors, the topological structure of the spaces $\operatorname{Ext}(X,Y)$ and its connection with the geometry of the spaces.