  Let X be a separable Banach space which admits a separating polynomial; in
particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded,
Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each
{\epsilon}>0, there exists an analytic function $g:X \rightarrow R$ with
$|g-f|<\epsilon$ and $||g'-f'||<\epsilon$.
