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.
We say that X is compactly embedded in Y, and write X ⊂⊂ Y or X ⋐ Y, if If Y is a Banach space, an equivalent definition is that the embedding operator (the identity) i : X → Y is a compact operator.
When applied to functional analysis, this version of compact embedding is usually used with Banach spaces of functions.
When an embedding is not compact, it may possess a related, but weaker, property of cocompactness.