  Let $X$ be a separable Banach space with a separating polynomial. We show
that there exists $C\geq 1$ (depending only on $X$) such that for every
Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\epsilon>0$, there
exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that
$|f(x)-g(x)|\leq \epsilon$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This
result is new even in the case when $X$ is a Hilbert space. Furthermore, in the
Hilbertian case we also show that $C$ can be assumed to be any number greater
than 1.
