Uniform nonextendability from nets

TL;DR

The paper demonstrates that in Banach spaces, certain Lipschitz functions defined on nets cannot be extended to uniformly continuous functions on the entire space, highlighting limitations in extension properties.

Contribution

It introduces the existence of specific Banach spaces and nets where Lipschitz functions cannot be extended uniformly continuously, revealing new limitations in extension theory.

Findings

Existence of Banach spaces with nonextendable Lipschitz functions

Construction of nets where extensions fail to be uniformly continuous

Highlights limitations in Lipschitz extension properties in Banach spaces

Abstract

It is shown that there exist Banach spaces $X,Y$, a $1$-net $\mathscr{N}$ of $X$ and a Lipschitz function $f:\mathscr{N}\to Y$ such that every $F:X\to Y$ that extends $f$ is not uniformly continuous.