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What is “local theory of Banach spaces”? (English) Zbl 0938.46011

One of the main purposes of the paper is to develop a point of view on the question: What is “local theory of Banach spaces”?
The author presents several answers that have already appeared in the literature and develops a new approach to understand this question using the new concepts of local representability of operators.
The paper should be of interest to experts interested in examining the concepts of the local theory of Banach spaces in high degree of generality.
Reviewer’s remark: V. D. Milman expressed the opinion [it seems that for the first time it will be published in the forthcoming survey: A. A. Giannopoulos and V. D. Milman, “Euclidean structure in finite-dimensional normed spaces”, in: Handbook on the geometry of Banach spaces, North-Holland, to appear] that the name “Asymptotic theory of finite-dimensional normed spaces”, would be a better name for the theory that is usually called the “Local theory of Banach spaces”. The reviewer supports this opinion of V. D. Milman for at least two reasons: 1) the word ‘local’ leads to wrong intuition; 2) finite-dimensional normed spaces were studied by H. Minkowski before the work of S. Banach.

MSC:

46B07 Local theory of Banach spaces
46B08 Ultraproduct techniques in Banach space theory
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