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Functional superanalysis. (English. Russian original) Zbl 0665.46031
Russ. Math. Surv. 43, No. 2, 103-137 (1988); translation from Usp. Mat. Nauk 43, No. 2(260), 87-114 (1988).
This is a quick introduction to and at the same time a survey of the functional analysis which may be described as the analysis of functions on superspaces over a commutative Banach superalgebra. This new branch of modern mathematics takes its origin in the papers by V. S. Vladimirov and I. V. Volovič [Teor. mat. fiz. 59, no. 1, 3-27 (1984; Zbl 0552.46023), 60, no. 2, 169-198 (1984; Zbl 0599.46068)] and the present author [ibid. 72, no. 3, 24-34 (1987); 66, no. 3, 339-349 (1986; Zbl 0622.35082)] and is of interest to mathematicians and, especially, to theoretical physicists.
The contents of the paper may be seen from the heading of chapters: Analysis on a superspace over a commutative Banach superalgebra, resp. commutative supermodule, Feynman and Gaussian distributions on a superspace, Hilbert superspace of states of a quantum system with bosonic and fermionic coordinates, pseudodifferential operators in the superanalysis, general theory of the superspace, possible generalizations and unsolved problems. The bibliography consists of 102 items, mainly in Russian.
Reviewer: J.Danes

MSC:
46G05 Derivatives of functions in infinite-dimensional spaces
45F10 Dual, triple, etc., integral and series equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
81S40 Path integrals in quantum mechanics
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46H99 Topological algebras, normed rings and algebras, Banach algebras
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