@article{Alvarado_Hajłasz_Malý_2023, title={A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces}, volume={48}, url={https://afm.journal.fi/article/view/127419}, DOI={10.54330/afm.127419}, abstractNote={<pre>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 \(p\in (1,\infty)\). 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 \(p\in (1,\infty)\).</pre> <p>&nbsp;</p>}, number={1}, journal={Annales Fennici Mathematici}, author={Alvarado, Ryan and Hajłasz, Piotr and Malý, Lukáš}, year={2023}, month={Mar.}, pages={255–275} }