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Compact embedding

Let (X, T) be a topological space, and let V and WbesubsetsofX. We say that Viscompactly embeddedinW, and write V ⊂⊂ W, if

• V ⊆ Cl(V) ⊆ Int(W), where Cl(V) denotes the closureofV, and Int(W) denotes the interiorofW; and
• Cl(V) is compact.

Definition (normed spaces)[edit]

Let X and Y be two normed vector spaces with norms ||•||_X and ||•||_Y respectively, and suppose that X ⊆ Y. We say that Xiscompactly embeddedinY, and write X ⊂⊂ Y, if

• Xiscontinuously embeddedinY; i.e., there is a constant C such that ||x||_Y ≤ C||x||_X for all xinX; and
• The embedding of X into Y is a compact operator: any bounded setinXistotally boundedinY, i.e. every sequence in such a bounded set has a subsequence that is Cauchy in the norm ||•||_Y.

IfY 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. 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.

References[edit]

• Adams, Robert A. (1975). Sobolev Spaces. Boston, MA: Academic Press. ISBN 978-0-12-044150-1..
• Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2..
• Renardy, M. & Rogers, R. C. (1992). An Introduction to Partial Differential Equations. Berlin: Springer-Verlag. ISBN 3-540-97952-2..