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

Abstract

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

17 pages

References in corpus (2)

• Geometry and nonlinear analysis
• Approximation by Lipschitz, analytic maps on certain Banach spaces

Cited by in corpus (1)

• Real analytic approximations which almost preserve Lipschitz constants of functions defined on the Hilbert space