Continuous embedding

In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space.

Definition

Let ${\displaystyle X}$  and ${\displaystyle Y}$  be two normed vector spaces, with norms ${\displaystyle \|\cdot \|_{X}}$  and ${\displaystyle \|\cdot \|_{Y}}$  respectively, such that ${\displaystyle X\subseteq Y}$ . If the inclusion map

${\displaystyle i:X\hookrightarrow Y:x\mapsto x}$ 

is continuous, i.e. if there exists a constant ${\displaystyle C\geq 0}$  such that

${\displaystyle \|x\|_{Y}\leq C\|x\|_{X}}$ 

for every ${\displaystyle x\in X}$ , then ${\displaystyle X}$  is continuously embedded in ${\displaystyle Y}$ .

See also

• Compactly embedded

Reference

• Rennardy, M., & Rogers, R.C. (1992). An Introduction to Partial Differential Equations. Springer-Verlag, Berlin. ISBN 3-540-97952-2.