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Structural theory of Banach spaces and its interplay with analysis and probability. (English) Zbl 0575.46010
Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 237-269 (1984).
[For the entire collection see Zbl 0553.00001.]
From the introduction: ”The purpose of this survey is to give some idea what the modern theory of Banach spaces is about.” This is organized as follows: after some historical remarks - where the modesty of the author has forbidden to include the name A. Pełczyński in the list of the mathematicians who contributed essentially to the renaissance of the theory - there is a first part on general Banach spaces and infinite- dimensional problems (e.g. Banach lattices, spaces without lattice structure, containment of \(\ell^ p)\). The second part deals with local theory from a more structural point of view (e.g. type and cotype, vector random series, absolutely summing and hilbertian operators, Dvoretzky’s theorem). The paper ends with a list of some omitted topics (e.g. aspects of the geometry of the unit ball of a Banach space) and an excellent bibliography with more than 200 entries.
Reviewer: P.Harmand

MSC:
46Bxx Normed linear spaces and Banach spaces; Banach lattices
46B20 Geometry and structure of normed linear spaces