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Let X be a separable Banach space which admits a separating polynomial; inparticular 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 $||g'-f'||<\epsilon$.

Approximation of functions and their derivatives by analytic maps on certain Banach spaces