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

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

2010-05-06

arxiv.org/abs/1005.1050v5

Mathematics

Functional Analysis

Updated version with a sharper result in the Hilbertian case. One thin tube is enough. Some misprints corrected

Scientific paper

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.

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