  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.
