Some problems in functional analysis inspired by Hahn-Banach type theorems. (English) Zbl 1317.46003

Sofi, M. A.

Ann. Funct. Anal. 5, No. 2, 1-29 (2014).

This interesting survey paper deals with various aspects of the Hahn-Banach extension theorem for Banach spaces. Among the topics covered by the author are the following: applications of the Hahn-Banach theorem in analysis, Hahn-Banach type theorems for operators, uniqueness of the Hahn-Banach extension, the Hahn-Banach theorem for bilinear mappings, properties around the Hahn-Banach theorem characterising finite-dimensional spaces, characterisations of Hilbert spaces by Hahn-Banach type properties, intersection properties of balls, extension of Lipschitz maps.
The author displays only few proofs, but points to the literature instead; the reference list, which contains a number of not so well-known papers, gives ample opportunity to pursue the subject matter further.
The paper would have benefitted from proper proof-reading and copy-editing; there are mathematical ambiguities, language slips, misspelt names and typographical imperfections that could have been avoided.

Reviewer: Dirk Werner (Berlin)

MSC:

46-02	Research exposition (monographs, survey articles) pertaining to functional analysis
46A20	Duality theory for topological vector spaces
46B03	Isomorphic theory (including renorming) of Banach spaces
46B04	Isometric theory of Banach spaces
46C15	Characterizations of Hilbert spaces
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)

Keywords:

Hahn-Banach extension theorem; injective Banach spaces; extension of Lipschitz maps; Hilbert-Schmidt spaces; intersection properties of balls