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. Several of the Sobolev embedding theorems are continuous embedding theorems.

Definition

Let X and Y be two normed vector spaces, with norms ||·||X and ||·||Y respectively, such that X ⊆ Y. If the inclusion map
is continuous, i.e. if there exists a constant C ≥ 0 such that
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 ↪ 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 morphisms are the continuous linear maps.

Examples