# Ehrling's lemma

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.

## Statement of the lemma

Let (X, ||·||X), (Y, ||·||Y) and (Z, ||·||Z) be three Banach spaces. Assume that:

Then, for every ε > 0, there exists a constant C(ε) such that, for all x ∈ X,

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

## Corollary (equivalent norms for Sobolev spaces)

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

## References

• Renardy, Michael; Rogers, Robert C. (1992). An Introduction to Partial Differential Equations. Berlin: Springer-Verlag. ISBN 978-3-540-97952-4.