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Some recent results concerning weak Asplund spaces. (English) Zbl 1059.46016

Authors’ abstract: This paper provides a gentle introduction to the study of weak Asplund spaces. It begins with a brief historical review of the development of weak Asplund spaces; starting from the earliest results concerning the differentiability of convex functions on \(\mathbb{R}\) through to the most recent developments concerning possible characterization of these spaces. Along the way, the classes of Gâteaux differentiability spaces, Stegall spaces and fragmentable spaces are introduced and their relationship with weak Asplund spaces reviewed. Following this, we summarize some of the most recent attempts at distinguishing the classes of weak Asplund spaces, Stegall spaces and fragmentable spaces. We conclude the paper by examining a class of topological spaces, namely the class of weak Asplund spaces, that may be useful in the problem of distinguishing the class of weak Asplund spaces from the class of Gâteaux differentiability spaces.

MSC:

46B26 Nonseparable Banach spaces
54C60 Set-valued maps in general topology
26E25 Set-valued functions
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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