Idempotent functional analysis: an algebraic approach

TL;DR

This paper reviews and develops an algebraic framework for Idempotent Functional Analysis, establishing analogs of classical theorems and concepts within an idempotent algebraic setting, inspired by Maslov's work.

Contribution

It introduces an algebraic approach to Idempotent Functional Analysis, generalizing classical theorems and defining key notions in an algebraic context.

Findings

Idempotent addition defined for arbitrary infinite sets of summands

Analogues of main theorems of linear functional analysis established

Characterization of linear functionals and scalar products in idempotent spaces

Abstract

In this paper we consider Idempotent Functional Analysis, an `abstract' version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a review of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of the traditional Functional Analysis and its idempotent version is discussed; this correspondence is similar to N. Bohr's correspondence principle in quantum theory. We present an algebraical approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraical 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 main theorems of linear functional analysis and results concerning the general form of a linear functional and scalar products in idempotent spaces.