A NOTE ON APPROXIMATION OF CONTINUOUS FUNCTIONS ON NORMED SPACES

Let X be a real separable normed space X admitting a separating polynomial. We prove that each continuous function from a subset A of X to a real Banach space can be uniformly approximated by restrictions to A of functions, which are analytic on open subsets of X. Also we prove that each continuous function to a complex Banach space from a complex separable normed space, admitting a separating *-polynomial, can be uniformly approximated by *-analytic functions.