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)
- V ⊆ Cl(V) ⊆ Int(W), where Cl(V) denotes the closure of V, and Int(W) denotes the interior of W; and
- Cl(V) is compact.
Definition (normed spaces)
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
- X is continuously embedded in Y; i.e., there is a constant C such that ||x||Y ≤ C||x||X for all x in X; and
- The embedding of X into Y is a compact operator: any bounded set in X is totally bounded in Y, i.e. every sequence in such a bounded set has a subsequence that is Cauchy in the norm ||•||Y.
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