Compact embedding

In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis. The notation for "${\displaystyle X}$ is compactly embedded in ${\displaystyle Y}$" is ${\displaystyle X\subset \subset Y}$, or ${\displaystyle X\Subset Y}$.

When used in functional analysis, compact embedding is usually about Banach spaces of functions.

Several of the Sobolev embedding theorems are compact embedding theorems.

When an embedding is not compact, it may possess a related, but weaker, property of cocompactness.