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

Abstract

Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\ge 1$ (depending only on $X$) such that for every Lipschitz function $f:X\to \mathbb{R}$, and every $\epsilon >0$, there exists a Lipschitz, real analytic function $g:X\to \mathbb{R}$ such that $|f(x)-g(x)|\le \epsilon$ and $\text{Lip}(g)\le C\text{Lip}(f)$. 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.

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

References in corpus (2)

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