Certain approximate properties of linear superpositions (Q1916569)

\(C(X)\) denotes the space of continuous real functions on some compact, \( B(X)\) -- the space of bounded real functions. The paper deals with uniform approximation in the spaces of functions of several variables, more precisely with an approximation of the so-called Vitushkin linear superpositions of continuous functions (subspace \(D)\) by the similar superpositions of bounded functions (subspce \(BD)\). A lemma on approximation of measures from an orthogonal to \(D\) subspace by measures with finite support is proved. Many results on the proximality of the subspace \(BD\) are proved under some conditions on the geometrical structure of variables \(X\) in \(B(X)\).