Homological Methods in Banach Space Theory

Contents
Preface
Preliminaries
1. Complemented Subspaces of Banach Spaces
	1.1 Banach and Quasi-Banach Spaces
	1.2 Complemented Subspaces
	1.3 Uncomplemented Subspaces
	1.4 Local Properties and Techniques
	1.5 The Dunford–Pettis, Grothendieck, Pełczyński and Rosenthal Properties
	1.6 C(K)-Spaces and Their Complemented Subspaces
	1.7 Sobczyk’s Theorem and Its Derivatives
	1.8 Notes and Remarks
2. The Language of Homology
	2.1 Exact Sequences of Quasi-Banach Spaces
	2.2 Basic Examples of Exact Sequences
	2.3 Topologically Exact Sequences
	2.4 Categorical Constructions for Absolute Beginners
	2.5 Pullback and Pushout
	2.6 Pushout and Exact Sequences
	2.7 Projective Presentations: the Universal Property of ????ₚ
	2.8 Pullbacks and Exact Sequences
	2.9 Injective Presentations: the Universal Property of ????_∞
	2.10 All about That Pullback/Pushout Diagram
	2.11 Diagonal and Parallel Principles
	2.12 Homological Constructions Appearing in Nature
	2.13 The Device
	2.14 Extension and Lifting of Operators
	2.15 Notes and Remarks
3. Quasilinear Maps
	3.1 An Introduction to Quasilinear Maps
	3.2 Quasilinear Maps in Action
	3.3 Quasilinear Maps versus Exact Sequences
	3.4 Local Convexity of Twisted Sums and ????-Spaces
	3.5 The Pullback and Pushout in Quasilinear Terms
	3.6 Spaces of Quasilinear Maps
	3.7 Homological Properties of ????ₚ and Lₚ When 0 < p ≤ 1
	3.8 Exact Sequences of Banach Spaces and Duality
	3.9 Different Versions of a Quasilinear Map
	3.10 Linearisation of Quasilinear Maps
	3.11 The Type of Twisted Sums
	3.12 A Glimpse of Centralizers
	3.13 Notes and Remarks
4. The Functor Ext and the Homology Sequences
	4.1 The Functor Ext
	4.2 The Homology Sequences
	4.3 Homology in Quasilinear Terms
	4.4 Alternative Constructions of Ext
	4.5 Topological Aspects of Ext
	4.6 Notes and Remarks
5. Local Methods in the Theory of Twisted Sums
	5.1 Local Splitting
	5.2 Uniform Boundedness Principles for Exact Sequences
	5.3 The Mysterious Role of the BAP
	5.4 Notes and Remarks
6 Fraïssé Limits by the Pound
	6.1 Fraïssé Classes and Fraïssé Sequences
	6.2 Almost Universal Disposition
	6.3 Almost Universal Complemented Disposition
	6.4 A Universal Operator on Gₚ
	6.5 Notes and Remarks
7. Extension of Operators, Isomorphisms and Isometries
	7.1 Operators: Extensible and UFO Spaces
	7.2 Isomorphisms: the Automorphic Space Problem
	7.3 Isometries: Universal Disposition
	7.4 Positions in Banach Spaces
	7.5 Notes and Remarks
8. Extension of C(K)-Valued Operators
	8.1 Zippin Selectors
	8.2 The Lindenstrauss–Pełczyński Theorem
	8.3 Kalton’s Approach to the C -Extension Property
	8.4 Sequence Spaces with the C -Extension Property
	8.5 C-Extensible Spaces
	8.6 The Dark Side of the Johnson–Zippin Theorem
	8.7 The Astounding Story behind the CCKY Problem
	8.8 Notes and Remarks
9. Singular Exact Sequences
	9.1 Basic Properties and Techniques
	9.2 Singular Quasilinear Maps
	9.3 Amalgamation Techniques
	9.4 Notes and Remarks
10. Back to Banach Space Theory
	10.1 Vector-Valued Versions of Sobczyk’s Theorem
	10.2 Polyhedral L_∞-Spaces
	10.3 Lipschitz and Uniformly Homeomorphic L_∞-Spaces
	10.4 Properties of Kernels of Quotient Maps on L₁ Spaces
	10.5 3-Space Problems
	10.6 Extension of L_∞-Valued Operators
	10.7 Kadec Spaces
	10.8 The Kalton–Peck Spaces
	10.9 The Properties of Z₂ Explained by Itself
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Index
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