On Hilbert extensions of Weierstrass' theorem with weights

In this paper we study the set of g-valued functions which can be approximated by g-valued continuous functions in the norm L-g(infinity) (I, w), where I subset of R is a compact interval, g is a separable real Hilbert space and w is a certain g-valued weakly measurable weight. Thus, we obtain a new extension of the celebrated Weierstrass approximation theorem.