Remarks on a lemma by Jacques-Louis Lions

Let Ω   be a bounded and connected open subset of R^N${\mathbb{R}}^N$ with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on one side of ∂Ω  . It is shown in this Note that a fundamental characterization of the space L²(Ω)$L^2(\Omega )$ due to Jacques-Louis Lions is in effect equivalent to a variety of other properties. One of the keys for establishing these equivalences is a specific “approximation lemma”, itself one of these equivalent properties.