Continuous embedding: Difference between revisions

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==Definition==

Let ''X'' and ''Y'' be two normed vector spaces, with norms ||&middot;·||_''X'' and ||&middot;·||_''Y'' respectively, such that <math>''X \subseteq ''&nbsp;⊆&nbsp;''Y</math>''. 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}_Y \leq C \| x \|_{X}_X</math>

for every ''x'' in ''X'', then ''X'' is said to be '''continuously embedded''' in ''Y''. Some authors use the hooked arrow "↪" to denote a continuous embedding, i.e. "''X''&nbsp;↪&nbsp;''Y''" means "''X'' and ''Y'' are normed spaces with ''X'' continuously embedded in ''Y''". This is a consistent use of notation from the point of view of the [[category of topological vector spaces]], in which the [[morphism]]s ("arrows") are the [[continuous linear map]]s.

for every ''x'' in ''X'', then ''X'' is said to be '''continuously embedded''' in ''Y''.

==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'''², where both spaces are given the Euclidean norm:

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

: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 Ω&nbsp;⊆&nbsp;'''R'''^''n'' be an [[open set|open]], [[bounded set|bounded]], [[Lipschitz ___domain]], and let 1&nbsp;≤&nbsp;''p''&nbsp;&lt;&nbsp;''n''. Set

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

:Then the Sobolev space ''W''^1,''p''(&Omega;;&nbsp;'''R''') is continuously embedded in the [[Lp space|''L''^''p'' space]] ''L''^{''p''^&lowast;}(&Omega;;&nbsp;'''R'''). In fact, for 1&nbsp;&le;&nbsp;''q''&nbsp;&lt;&nbsp;''p''^&lowast;, this embedding is [[compactly embedded|compact]]. The optimal constant ''C'' will depend upon the geometry of the ___domain &Omega;.

* Infinite-dimensional spaces also offer examples of ''discontinuous'' embeddings. For example, consider

::<math>X = Y = C^0 ([0, 1]; \mathbf{R}),</math>

:the space of continuous real-valued functions defined on the unit interval, but equip ''X'' with the ''L''¹ norm and ''Y'' with the [[supremum norm]]. For ''n''&nbsp;&isin;&nbsp;'''N''', let ''f''_''n'' be the [[continuous function|continuous]], [[piecewise linear function]] given by

::<math>f_n (x) = \begin{cases} - n^2 x + n , & 0 \leq x \leq \tfrac 1 n; \\ 0, & \text{otherwise.} \end{cases}</math>

:Then, for every ''n'', ||''f''_''n''||_''Y''&nbsp;=&nbsp;||''f''_''n''||_∞&nbsp;=&nbsp;''n'', but

::<math>\| f_n \|_{L^1} = \int_0^1 | f_n (x) | \, \mathrm{d} x = \frac1{2}.</math>

:Hence, no constant ''C'' can be found such that ||''f''_''n''||_''Y''&nbsp;&le;&nbsp;''C''||''f''_''n''||_''X'', and so the embedding of ''X'' into ''Y'' is discontinuous.

==See also==

* [[CompactlyCompact embeddedembedding]]

==ReferenceReferences==

* {{cite book | authorauthor1=RennardyRenardy, M., &|author2= Rogers, R.C. |name-list-style=amp | title=An Introduction to Partial Differential Equations | publisher=Springer-Verlag, Berlin | year=1992 | idisbn=ISBN 3-540-97952-2 }}

[[Category:Functional analysis]]