On Hilbert extensions of Weierstrass' theorem with weights

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