mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 (''X'', ''T'') be a topological space, and let ''V'' and ''W'' be

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of ''X''. We say that ''V'' is compactly embedded in ''W'', and write ''V'' ⊂⊂ ''W'', if * ''V'' ⊆ Cl(''V'') ⊆ Int(''W''), where Cl(''V'') denotes the closure of ''V'', and Int(''W'') denotes the

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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 :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mat ...

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''. If ''Y'' is a

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

, 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

* . * . * {{cite book , author1=Renardy, M. , author2= Rogers, R. C. , name-list-style=amp , title=An Introduction to Partial Differential Equations , location=Berlin , publisher=Springer-Verlag , year=1992 , isbn=3-540-97952-2 . Compactness (mathematics) Functional analysis General topology