Idempotent Functional Analysis : an Algebraical Approach

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.