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| In [[mathematics]], '''Ehrling's lemma''' is a result concerning [[Banach space]]s. It is often used in [[functional analysis]] to demonstrate the [[norm (mathematics)#Properties|equivalence]] of certain [[norm (mathematics)|norms]] on [[Sobolev space]]s.
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| ==Statement of the lemma==
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| Let (''X'', ||·||<sub>''X''</sub>), (''Y'', ||·||<sub>''Y''</sub>) and (''Z'', ||·||<sub>''Z''</sub>) be three Banach spaces. Assume that:
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| * ''X'' is [[compactly embedded]] in ''Y'': i.e. ''X'' ⊆ ''Y'' and every ||·||<sub>''X''</sub>-[[bounded function|bounded]] [[sequence]] in ''X'' has a [[subsequence]] that is ||·||<sub>''Y''</sub>-[[Limit (mathematics)|convergent]]; and
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| * ''Y'' is [[continuously embedded]] in ''Z'': i.e. ''Y'' ⊆ ''Z'' and there is a constant ''k'' so that ||''y''||<sub>''Z''</sub> ≤ ''k''||''y''||<sub>''Y''</sub> for every ''y'' ∈ ''Y''.
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| Then, for every ''ε'' > 0, there exists a constant ''C''(''ε'') such that, for all ''x'' ∈ ''X'',
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| :<math>\| x \|_{Y} \leq \varepsilon \| x \|_{X} + C(\varepsilon) \| x \|_{Z}</math>
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| ==Corollary (equivalent norms for Sobolev spaces)== | |
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| Let Ω ⊂ '''R'''<sup>''n''</sup> be [[open set|open]] and [[bounded set|bounded]], and let ''k'' ∈ '''N'''. Suppose that the Sobolev space ''H''<sup>''k''</sup>(Ω) is compactly embedded in ''H''<sup>''k''−1</sup>(Ω). Then the following two norms on ''H''<sup>''k''</sup>(Ω) are equivalent:
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| :<math>\| \cdot \| : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \| u \| := \sqrt{\sum_{| \alpha | \leq k} \| \mathrm{D}^{\alpha} u \|_{L^{2} (\Omega)}^{2}}</math> | |
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| and
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| :<math>\| \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}}.</math>
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| For the subspace of ''H''<sup>''k''</sup>(Ω) consisting of those Sobolev functions with [[trace operator|zero trace]] (those that are "zero on the boundary" of Ω), the ''L''<sup>1</sup> norm of ''u'' can be left out to yield another equivalent norm.
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| ==References==
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| * {{cite book
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| | last1 = Renardy
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| | first1 = Michael
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| | last2 = Rogers
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| | first2 = Robert C.
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| | title = An Introduction to Partial Differential Equations
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| | publisher = Springer-Verlag
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| | location = Berlin
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| | year=1992
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| | isbn=978-3-540-97952-4
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| }}
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| [[Category:Banach spaces]]
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| [[Category:Sobolev spaces]]
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| [[Category:Lemmas]]
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| {{mathanalysis-stub}}
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