This publication offers with difficulties of approximation of continuing or bounded capabilities of numerous variables through linear superposition of features which are from an analogous classification and feature fewer variables. the most subject is the gap of linear superpositions $D$ regarded as a subspace of the gap of continuing features $C(X)$ on a compact area $X$. Such homes as density of $D$ in $C(X)$, its closedness, proximality, and so on. are studied in nice element. The method of those and different difficulties in accordance with duality and the Hahn-Banach theorem is emphasised. additionally, massive awareness is given to the dialogue of the Diliberto-Straus set of rules for locating the simplest approximation of a given functionality by means of linear superpositions.
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