Ehrling's lemma

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In mathematics, Ehrling's lemma is a result concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev spaces.

[edit] 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,

$\| x \|_{Y} \leq \varepsilon \| x \|_{X} + C(\varepsilon) \| x \|_{Z}$

[edit] Corollary (equivalent norms for Sobolev spaces)

Let Ω ⊂ Rⁿ be open and bounded, and let k ∈ N. Suppose that the Sobolev space H^k(Ω) is compactly embedded in H^k−1(Ω). Then the following two norms on H^k(Ω) are equivalent:

$\| \cdot \| : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \| u \| := \sqrt{\sum_{| \alpha | \leq k} \| \mathrm{D}^{\alpha} u \|_{L^{2} (\Omega)}^{2}}$

and

$\| \cdot \|' : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \| u \|' := \sqrt{\| u \|_{L^{1} (\Omega)}^{2} + \sum_{| \alpha | = k} \| \mathrm{D}^{\alpha} u \|_{L^{2} (\Omega)}^{2}}.$

For the subspace of H^k(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.