Kirszbraun theorem

A theorem in real analysis, which states that, if $E\subset \mathbb R^n$, then any Lipschitz function $f: E \to \mathbb R^m$ can be extended to the whole $\mathbb R^n$ keeping the Lipschitz constant of the original function. In the case $m=1$ the theorem is rather straightforward, since one such extension is given by \[ \tilde{f} (x) := \inf_{y\in E} f (x) + {\rm Lip (f)} |x-y|\, . \] The general case $m>1$ is instead rather complicated. Note that a Lipschitz extension with a non-optimal costant can be easily achieved using the formula above for each component of the vector function.

The theorem remains valid if both $\mathbb R^n$ and $\mathbb R^m$ are replaced by general Hilbert spaces $H_1$ and $H_2$. When $H_1$ is not separable such extension requires some form of the Axiom of choice. The conclusion of Kirszbraun's theorem is instead false as soon as we leave the Hilbert setting (for instance replacing either of the $H_i$'s with a Banach space).