Albiac F., Kalton N.J. Topics in Banach Space Theory

2nd Edition. — Springer International Publishing Switzerland, 2016. — 512 p. — (Graduate Texts in Mathematics 233) — ISBN: 978-3-319-31555-3.Ten years ago, Fernando Albiac and Nigel Kalton completed their book Topics in Banach Space Theory. Sometimes, the mathematical community recognizes the true value of a fine work. Fortunately, that is what happened with this book, which has been successful and influential, to such an extent that a second edition is necessary. It is appropriate to ponder this success. As explained in the preface to the first edition, this book grew out of two graduate courses delivered by Nigel Kalton at the University of Missouri–Colombia. It therefore reflects the enlightening vision of a master in this field: a cascade falling from so high is a powerful force, and a beautiful sight. But after the cascade, the stream and the river flow. Once the work was put into motion, much remained to be done. And the authors’ wisdom led them to gather their “topics in Banach space theory” according to the following rules: choose proofs that are accessible to graduate students, pick positive results in the various fields where Banach space is the operative word, make every attempt to address the largest possible audience, explain concisely counterexamples without fully displaying the technicalities. Their efforts resulted in a well-balanced, instructive, and reader-friendly masterpiece.Bases and Basic Sequences
The Classical Sequence Spaces
Special Types of Bases
Banach Spaces of Continuous Functions
L1/-Spaces and C.K/-Spaces
The Spaces Lp for 1 p < 1
Factorization Theory
Absolutely Summing Operators
Perfectly Homogeneous Bases and Their Applications
Greedy-Type Bases
lp-Subspaces of Banach Spaces
Finite Representability of `p-Spaces
An Introduction to Local Theory
Nonlinear Geometry of Banach Spaces
Important Examples of Banach SpacesNormed Spaces and Operators
Elementary Hilbert Space Theory
Duality in Lp./: Results Related to Hölder’s Inequality
Main Features of Finite-Dimensional Spaces
Cornerstone Theorems of Functional Analysis
Convex Sets and Extreme Points
The Weak Topologies
Weak Compactness of Sets and Operators
Basic Probability in Use
Generalities on Ultraproducts
The Bochner Integral Abridged