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

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