By defining idempotent semigroup, idempotent semiring, idempotent semimodule and idempotent space, an algebraic approach to idempotent functional analysis is presented in the paper. The idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of linear functionals (Theorem 1), Hahn-Banach Theorem (Theorem 2), Banach-Steinhaus Theorem (Proposition 2), Closed-Graph Theorem (Proposition 3) and Riesz-Fischer Theorem (Theorem 3) are discussed in this paper.
It has been remarked that using the completion procedures, one can extend all the results obtained to the case of incomplete semirings, spaces and semimodules.