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Idempotent functional analysis: An algebraic approach. (English. Russian original) Zbl 1017.46034
Math. Notes 69, No. 5, 696-729 (2001); translation from Mat. Zametki 69, No. 5, 758-797 (2001).
Summary: This paper is devoted to idempotent functional analysis, which is an “abstract” version of idempotent analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of idempotent analysis. The correspondence between concepts and theorems of traditional functional analysis and its idempotent version is discussed in the spirit of N. Bohr’s correspondence principle in quantum theory. We present an algebraic approach to idempotent functional analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.

MSC:
46S99 Other (nonclassical) types of functional analysis
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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