Ehrling's lemma

In mathematics, Ehrling's lemma, also known as Lions' lemma, is a result concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev spaces. It was named after Gunnar Ehrling.

Statement of the lemma

Let, and be three Banach spaces. Assume that:X is compactly embedded in Y: i.e. X ⊆ Y and every ||⋅||_X-bounded sequence in X has a subsequence that is ||⋅||_Y-convergent; andY is continuously embedded in Z: i.e. Y ⊆ Z and there is a constant k so that ||y||_Z ≤ k||y||_Y for every y ∈ Y.
Then, for every ε > 0, there exists a constant C such that, for all x ∈ X,

Let Ω ⊂ Rⁿ be open and bounded, and let k ∈ N. Suppose that the Sobolev space H^k is compactly embedded in H^k−1. Then the following two norms on H^k are equivalent:
and
For the subspace of H^k consisting of those Sobolev functions with zero trace, the L² norm of u can be left out to yield another equivalent norm.