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The Hahn-Banach theorem surveyed. (English) Zbl 0808.46003

This paper dedicated to the 100th birthday of Stefan Banach presents briefly and without proofs some important considerations on the famous Hahn-Banach Theorem, one of the fundamental principles of functional analysis, known in several different equivalent formulations.
In the first part of the paper, the author discusses the celebrated proofs due to Banach, Luxemburg, Nachbin, Mazur, and the extension to complex spaces given by Sukhomlinov and Bohnenblust-Sobczyk. In the next, the author points out important versions and generalizations of the Hahn- Banach Theorem. Also, several important problems in functional analysis generated by the Hahn-Banach theorem are extensively commented: constructive analysis and unique extensions, sandwich theorems, Mazur- Orlicz theorem, simultaneous Hahn-Banach extensions, the interpolation property, injective Banach spaces and injective Banach lattices, invariant extensions, Banach limits, amenability property.
The last part of this paper presents some special abstract structures in which a Hahn-Banach theorem is available. The paper contains a bibliography of 351 items, effectively mentioned in text. The writing is clear and to the point, and, contrary to what appears to be a current trend, historical interpretations and comments are supported by quotations, arguments and specific references.

MSC:

46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
03E25 Axiom of choice and related propositions
46M10 Projective and injective objects in functional analysis
46S20 Nonstandard functional analysis
46B20 Geometry and structure of normed linear spaces
46B42 Banach lattices
46S30 Constructive functional analysis
46C15 Characterizations of Hilbert spaces
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