A simple proof of reflexivity and separability of N1,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-Poincare inequality, then the N1,P(X) Sobolev space is reflexive and separable whenever p is an element of (1, infinity). We also prove separability of the space when p = 1. Our proof is based on a straightforward construction of an equivalent norm on N1,P(X), p is an element of [1, infinity), that is uniformly convex when p is an element of (1, infinity). Finally, we explicitly construct a functional that is pointwise comparable to the minimal p-weak upper gradient, when p is an element of (1, infinity).