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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'' = '''R'' into the plane ''Y'' = '''R'''<sup>2</sup>, 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''||<sub>''X''</sub> = ||''x''||<sub>''Y''</sub> for every real number ''X''. Clearly, the optimal choice of constant ''C'' is ''C'' = 1.
* An infinite-dimensional example of a continuous embedding is given by the [[Rellich-Kondrachov theorem]]: let Ω ⊆ '''R'''<sup>''n''</sup> be an [[open set|open]], [[bounded set|bounded]], [[Lipschitz ___domain]], and let 1 ≤ ''p'' < ''n''. Set
::<math>p^{*} = \frac{n p}{n - p}.</math>
:Then the Sobolev space ''W''<sup>1,''p''</sup>(Ω; '''R''') is continuously embedded in the [[Lp space|''L''<sup>''p''</sup> space]] ''L''<sup>''p''<sup>∗</sup></sup>(Ω; '''R'''). In fact, for 1 ≤ ''q'' < ''p''<sup>∗</sup>, this embedding is [[compactly embedded|compact]]. The optimal constant ''C'' will depend upon the geometry of the ___domain Ω.
* 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''<sup>1</sup> norm and ''Y'' with the [[supremum norm]]. For ''n'' ∈ '''N''', let ''f''<sub>''n''</sub> 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, & \mbox{otherwise.} \end{cases}</math>
:Then, for every ''n'', ||''f''<sub>''n''</sub>||<sub>''Y''</sub> = ||''f''<sub>''n''</sub>||<sub>∞</sub> = ''n'', but
::<math>\| f \|_{L^{1}} = \int_{0}^{1} | f_{n} (x) | \, \mathrm{d} x = \frac1{2}.</math>
:Hence, no constant ''C'' can be found such that ||''f''<sub>''n''</sub>||<sub>''Y''</sub> ≤ ''C''||''f''<sub>''n''</sub>||<sub>''X''</sub>, and so the embedding of ''X'' into ''Y'' is discontinuous.
==See also==
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