Statement of the lemma
Let (X, ||·||X), (Y, ||·||Y) and (Z, ||·||Z) 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; and
- Y 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(ε) such that, for all x ∈ X,
Corollary (equivalent norms for Sobolev spaces)
For the subspace of Hk(Ω) consisting of those Sobolev functions with zero trace (those that are "zero on the boundary" of Ω), the L1 norm of u can be left out to yield another equivalent norm.
- Renardy, Michael; Rogers, Robert C. (1992). An Introduction to Partial Differential Equations. Berlin: Springer-Verlag. ISBN 978-3-540-97952-4.
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