A simple proof of reflexivity and separability of N^1,p Sobolev spaces - Alvarado, Ryan; Hajłasz, Piotr; Malý, Lukáš (Annales Fennici Mathematici, 2023)
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 \(p(1,∞)\). We also prove separability of the space when \(p=1\). Our proof is based on a straightforward construction of an equivalent norm on \(N^1,p(X)\), \(p[1,∞)\), that is uniformly convex when \(p(1,∞)\). Finally, we explicitly construct a functional that is pointwise comparable to the minimal \(p\)-weak upper gradient, when \(p(1,∞)\).