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Three-space problems in Banach space theory. (English) Zbl 0914.46015

Lecture Notes in Mathematics. 1667. Berlin: Springer. xii, 267 p. (1997).
Let \(P\) be a subclass in the class of all Banach spaces. The three-space problem for \(P\) is stated in the following way.
Problem. Let \(Y\) be a subspace of a Banach space \(X\). Is it true that \(X\) is in \(P\) provided \(Y\) and the quotient \(X/Y\) are in \(P\)?
The book contains a survey of results on such problems for many different classes (=properties). The most useful approaches are described in detail.
At the end of the book there is a summary of results on three-space problems for 160 different classes (=properties). The book contains a useful bibliography.
As a result the book is a useful source of information for all interested in Banach space theory.
Some additions to bibliography.
1. The forthcoming paper: Yu. Brudnyi and N. J. Kalton, Polynomial approximation on subsets of \(\mathbb{R}^n\), contains important information on three-space problems.
2. The paper: N. J. Kalton and M. I. Ostrovskii, Distances between Banach spaces, Forum Math., 11, 17-48 (1999), contains many results on the topics touched on in Appendix 2.10.
3. It is worthwhile to mention that the main step in the proof of Theorem 2.10.a goes back to A. Douady, see pp. 15-16 of his paper: “Le problème des modules pour les sous espaces d’un espace analytique donnée”, Ann. Inst. Fourier 15, 1-94 (1966).
Reviewer’s remark: There is a confusion between different notions of stability in Appendix 2.10: the classes of all \({\mathcal L}_1\)-spaces and \({\mathcal L}_\infty\)-spaces are not stable in the sense defined on p. 72, they are open only. (The reason for this confusion is that R. Janz uses another terminology in his paper).

MSC:

46B20 Geometry and structure of normed linear spaces
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46B03 Isomorphic theory (including renorming) of Banach spaces
46B08 Ultraproduct techniques in Banach space theory
46B10 Duality and reflexivity in normed linear and Banach spaces
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