Banach spaces and topology. (English) Zbl 0584.46007

Handbook of set-theoretic topology, 1045-1142 (1984).
[For the entire collection see Zbl 0546.00022.]
The present survey on Banach spaces and topology consists of eight sections which are:
1. Rosenthal’s theorem for isomorphic embedding of \(\ell^ 1\) into Banach spaces.
2. Calibers, independent families of compact spaces K and universal isomorphic embedding of \(\ell^ 1_{\alpha}\) into C(K).
3. Isomorphic embeddings of \(\ell^ 1_{\alpha}\) in Banach spaces X and independent families on the dual unit ball \(S_{X^*}.\)
4. Large subsets of \(L^{\infty}(\mu)\) far apart in \(L_ 1\)-norm and Pelczynski conjecture
5. The Kunen-Haydon-Talagrand example.
6. Corson-compact spaces and subclasses - applications to Banach spaces.
Kunen’s example of an S-space and Banach spaces.
8. Fixed points of contractions in weakly compact convex subsets of Banach spaces.
Th author uses methods and results of infinitary combinatorics, compactness and diagonal arguments and ultraproducts. These ideas are exposed in the first four sections. The example of Kunen-Haydon-Talagrand which assumes CH is described in Section 5. Section 6 deals with various modifications of compactness, as, e.g., compactness in the sense of Gul’ko, Corson, Eberlein and others. Section 1 presents the Kunen’s example of an S-space and finally the Section 8 concerns fixed points of contractions on weakly compact convex sets.
Reviewer: L.Janos


46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
03C20 Ultraproducts and related constructions
03E50 Continuum hypothesis and Martin’s axiom


Zbl 0546.00022