Gâteaux differentiability of convex functions and topology. Weak Asplund spaces. (English) Zbl 0883.46011

Fabian, Marián J.

Canadian Mathematical Society Series of Monographs and Advanced Texts. New York, NY: Wiley. xi, 180 p. (1997).

A Banach space \(X\) is weak Asplund provided that each convex continuous function over \(X\) is Gâteaux differentiable on some dense \(G_{\delta}\) subset of \(X\). The classical Mazur’s theorem claims that every separable Banach space is weak Asplund. Thorough research into weak Asplund space is the topic of the book. The author exposes all relevant constructions and types of spaces. Among them are GSG spaces, WCG spaces, weakly countably determined spaces, preduals to spaces of Stegall’s class, etc. The exposition naturally involves the topological technique of Eberlein, Gul’ko, and Radon-Nikodým compacta.
The book consists of eight chapters:
1. Canonical examples of Weak Asplund Spaces.
2. Properties of Gâteuax Differentiability Spaces and Weak Asplund spaces.
3. Stegall’s Class.
4. Two More Concrete Classes of Banach Spaces that lie in \(\widetilde S\).
5. Fragmentability.
6. “Long Sequences” of Linear Projections.
7. Vašák Spaces and Gul’ko Compacta.
3. A Characterization of WCG Spaces and of Eberlein Compacta.
The book is supplemented with a short list of open questions. It also contains many apt remarks, a detailed index and a good selection of references. On the whole, the book is a welcome addition to the modern treasure trove of research monographs on the geometry of Banach spaces and convex analysis.

Reviewer: S.S.Kutateladze (Novosibirsk)

MSC:

46B20	Geometry and structure of normed linear spaces
46-02	Research exposition (monographs, survey articles) pertaining to functional analysis
46A50	Compactness in topological linear spaces; angelic spaces, etc.
46A55	Convex sets in topological linear spaces; Choquet theory
46G05	Derivatives of functions in infinite-dimensional spaces
46B10	Duality and reflexivity in normed linear and Banach spaces
46B22	Radon-Nikodým, Kreĭn-Milman and related properties

Keywords:

Asplund spaces; WCG spaces; fragmentability; Eberlein compacta; Radon-Nikodým compacta; weak Asplund; convec continuous function; Gâteaux differentiable; Mazur’s theorem; GSG spaces; weakly countably determined spaces; preduals; Stegall’s class; long sequences; Vašák spaces; Gul’ko compacta