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 In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

and

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

Definition (topological spaces)

Let (''X'', ''T'') be a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

, 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

interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...

of ''W''; and * Cl(''V'') is

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Definition (normed spaces)

Let ''X'' and ''Y'' be two

normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...

s 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 mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both def ...

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 In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...

: 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 topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...

in ''Y'', i.e. every

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

in such a bounded set has a

subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...

that is

Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...

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

. When applied to functional analysis, this version of compact embedding is usually used with

Banach spaces 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 ...

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