Book contents
- Frontmatter
- Contents
- Preface
- Preliminaries
- 1 Complemented Subspaces of Banach Spaces
- 2 The Language of Homology
- 3 Quasilinear Maps
- 4 The Functor Ext and the Homology Sequences
- 5 Local Methods in the Theory of Twisted Sums
- 6 Fraïssé Limits by the Pound
- 7 Extension of Operators, Isomorphisms and Isometries
- 8 Extension of C(K)-Valued Operators
- 9 Singular Exact Sequences
- 10 Back to Banach Space Theory
- Bibliography
- Index
10 - Back to Banach Space Theory
Published online by Cambridge University Press: 19 January 2023
- Frontmatter
- Contents
- Preface
- Preliminaries
- 1 Complemented Subspaces of Banach Spaces
- 2 The Language of Homology
- 3 Quasilinear Maps
- 4 The Functor Ext and the Homology Sequences
- 5 Local Methods in the Theory of Twisted Sums
- 6 Fraïssé Limits by the Pound
- 7 Extension of Operators, Isomorphisms and Isometries
- 8 Extension of C(K)-Valued Operators
- 9 Singular Exact Sequences
- 10 Back to Banach Space Theory
- Bibliography
- Index
Summary
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.
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- Information
- Homological Methods in Banach Space Theory , pp. 468 - 520Publisher: Cambridge University PressPrint publication year: 2023