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arXiv · 1005.1050

Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces

D. Azagra·R. Fry·L. Keener
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Abstract

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 $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ 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.

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D. Azagra, R. Fry, L. Keener. 2010-12-31. Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces. https://arxiv.org/abs/1005.1050

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