Continuous embedding: Difference between revisions

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==Definition==

Let <math>''X</math>'' and <math>''Y</math>'' be two normed vector spaces, with norms ||&middot;||<mathsub>\| \cdot \|_{''X}''</mathsub> and <math>\| \cdot \|_{&middot;||<sub>''Y}''</mathsub> respectively, such that <math>''X \subseteq ''&nbsp;&sube;&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 <math>''C \geq ''&nbsp;&ge;&nbsp;0</math> such that

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

for every <math>''x'' \in ''X</math>'', then <math>''X</math>'' is said to be '''continuously embedded''' in <math>''Y</math>''.

==See also==