Polynomial approximations and white noise integrals

In this paper a problem concerning polynomial approximability of random variables in the weak distribution framework is considered. The result, which can be viewed as a stochastic version of the Weierstrass theorem in Hilbert spaces, consists in a theorem which guarantees such polynomial approximation provided that a special continuity hypotheses holds for the random variable considered. The theory is useful for the polynomial approximation of the white noise integrals.