Continuous embedding: Difference between revisions

Revision as of 14:31, 8 October 2007

Revision as of 18:49, 10 October 2007

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==Definition==

Let ''X'' and ''Y'' be two normed vector spaces, with norms ||··||_''X'' and ||··||_''Y'' respectively, such that ''X'' ⊆⊆ ''Y''. If the [[identity function|inclusion map (identity function)]]

:<math>i : X \hookrightarrow Y : x \mapsto x</math>

is continuous, i.e. if there exists a constant ''C''&nbsp;&ge;≥&nbsp;0 such that

:<math>\| x \|_{Y} \leq C \| x \|_{X}</math>

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==Examples==

* A finite-dimensional example of a continuous embedding is given by a natural embedding of the [[real line]] ''X''&nbsp;=&nbsp;'''R'' into the plane ''Y''&nbsp;=&nbsp;'''R'''²², where both spaces are given the Euclidean norm:

::<math>i : \mathbf{R} \to \mathbf{R}^{2} : x \mapsto (x, 0)</math>

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:In this case, ||''x''||_''X''&nbsp;=&nbsp;||''x''||_''Y'' for every real number ''X''. Clearly, the optimal choice of constant ''C'' is ''C''&nbsp;=&nbsp;1.

* An infinite-dimensional example of a continuous embedding is given by the [[Rellich-Kondrachov theorem]]: let &Omega;Ω&nbsp;&sube;⊆&nbsp;'''R'''^''n'' be an [[open set|open]], [[bounded set|bounded]], [[Lipschitz ___domain]], and let 1&nbsp;&le;≤&nbsp;''p''&nbsp;&lt;&nbsp;''n''. Set

::<math>p^{*} = \frac{n p}{n - p}.</math>

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* [[Compactly embedded]]

==ReferenceReferences==

* {{cite book | author=Rennardy, M., & Rogers, R.C. | title=An Introduction to Partial Differential Equations | publisher=Springer-Verlag, Berlin | year=1992 | id=ISBN 3-540-97952-2 }}