Continuous embedding: Difference between revisions

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

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

:<math>\| x \|_Y \leq C \| x \|_X</math>

for every ''x'' in ''X'', then ''X'' is said to be '''continuously embedded''' in ''Y''. Some authors use the hooked arrow &ldquo;&#x21aa;&rdquo;"↪" to denote a continuous embedding, i.e. &ldquo;"''X''&nbsp;&#x21aa;↪&nbsp;''Y''&rdquo;" means &ldquo;"''X'' and ''Y'' are normed spaces with ''X'' continuously embedded in ''Y''&rdquo;". This is a consistent use of notation from the point of view of the [[category of topological vector spaces]], in which the [[morphism]]s (&ldquo;"arrows&rdquo;") are the [[continuous linear map]]s.

==Examples==

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::<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''||_&infin;∞&nbsp;=&nbsp;''n'', but

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

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==References==

* {{cite book |author1=RennardyRenardy, M. |author2= Rogers, R.C. |lastname-authorlist-ampstyle=yesamp | title=An Introduction to Partial Differential Equations | publisher=Springer-Verlag, Berlin | year=1992 | isbn=3-540-97952-2 }}

[[Category:Functional analysis]]