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 $(X, \| \cdot \|_{X})$ , $(Y, \| \cdot \|_{Y})$ and $(Z, \| \cdot \|_{Z})$ be three Banach spaces. Assume that:

$X$ is compactly embedded in $Y$ : i.e. $X \subseteq Y$ and every $\| \cdot \|_{X}$ -bounded sequence in $X$ has a subsequence that is $\| \cdot \|_{Y}$ -convergent; and
$Y$ is continuously embedded in $Z$ : i.e. $Y \subseteq Z$ and there is a constant $k$ with $\| y \|_{Z} \leq k \| y \|_{Y}$ for every $y \in Y$ .

Then, for every $\varepsilon > 0$ , there exists a constant $C(\varepsilon)$ such that, for all $x \in X$ ,

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

[edit] Corollary (equivalent norms for Sobolev spaces)

Let $\Omega \subsetneq \mathbb{R}^{n}$ be open and bounded, and let $k \in \mathbb{N}$ . 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:

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

and

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