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.

Definition (topological spaces)

Let be a topological space, and let V and W be subsets of X. We say that V is compactly embedded in W, and write V ⊂⊂ W, if
Let X and Y be two normed vector spaces with norms ||•||X and ||•||Y respectively, and suppose that X ⊆ Y. We say that X is compactly embedded in Y, and write X ⊂⊂ Y, if
If Y is a Banach space, an equivalent definition is that the embedding operator 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. 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.