A simple proof of reflexivity and separability of $N^{1,p}$ Sobolev spaces

TL;DR

This paper provides an elementary proof that Sobolev spaces on metric-measure spaces are reflexive and separable under certain conditions, using a straightforward norm construction and explicit functionals.

Contribution

It introduces a simple, constructive proof of reflexivity and separability of $N^{1,p}$ Sobolev spaces, including explicit norms and functionals.

Findings

Sobolev space $N^{1,p}(X)$ is reflexive for $p eq 1$

Space $N^{1,1}(X)$ is separable

Constructs an explicit functional comparable to the minimal $p$-weak upper gradient

Abstract

We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space $X$ supports a $p$-Poincar\'e inequality, then the $N^{1,p}(X)$ Sobolev space is reflexive and separable whenever $p\in (1,\infty)$. We also prove separability of the space when $p=1$. Our proof is based on a straightforward construction of an equivalent norm on $N^{1,p}(X)$, $p\in [1,\infty)$, that is uniformly convex when $p\in (1,\infty)$. Finally, we explicitly construct a functional that is pointwise comparable to the minimal $p$-weak upper gradient, when $p\in (1,\infty)$.