Inmathematics, the notion of beingcompactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to generaltopology andfunctional analysis. The notation for "${\displaystyle X}$ is compactly embedded in${\displaystyle Y}$" is${\displaystyle X\subset \subset Y}$, or${\displaystyle X⋐Y}$.

When used in functional analysis, compact embedding is usually aboutBanach spaces of functions.

Several of theSobolev embedding theorems are compact embedding theorems.

When an embedding is not compact, it may possess a related, but weaker, property ofcocompactness.

Let${\displaystyle X}$ be atopological space, and let${\displaystyle V}$ and${\displaystyle W}$ besubsets of${\displaystyle X}$. We say that${\displaystyle V}$ iscompactly embedded in${\displaystyle W}$ if

• ${\displaystyle V\subseteq \mathrm{Cl}(V)\subseteq \mathrm{Int}(W)}$, where${\displaystyle \mathrm{Cl}(V)}$ denotes theclosure of${\displaystyle V}$, and${\displaystyle \mathrm{Int}(W)}$ denotes theinterior of${\displaystyle W}$; and
• ${\displaystyle \mathrm{Cl}(V)}$ iscompact.

Equivalently, it states that there exists some compact set${\displaystyle K}$, such that${\displaystyle V\subseteq K\subseteq \mathrm{Int}(W)}$.

Let${\displaystyle X}$ and${\displaystyle Y}$ be twonormed vector spaces with norms${\displaystyle \Vert \cdot {\Vert }_X}$ and${\displaystyle \Vert \cdot {\Vert }_Y}$ respectively, and suppose that${\displaystyle X\subseteq Y}$. We say that${\displaystyle X}$ iscompactly embedded in${\displaystyle Y}$, if

• ${\displaystyle X}$ iscontinuously embedded in${\displaystyle Y}$; i.e., there is a constant${\displaystyle C}$ such that${\displaystyle \Vert x{\Vert }_Y\le C\Vert x{\Vert }_X}$ for all${\displaystyle x}$ in${\displaystyle X}$; and
• The embedding of${\displaystyle X}$ into${\displaystyle Y}$ is acompact operator: anybounded set in${\displaystyle X}$ istotally bounded in${\displaystyle Y}$, i.e. everysequence in such a bounded set has asubsequence that isCauchy in the norm${\displaystyle \Vert \cdot {\Vert }_Y}$.

If${\displaystyle Y}$ is aBanach space, an equivalent definition is that the embedding operator (the identity)${\displaystyle i:X\to Y}$ is acompact operator.

• 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.

• Compactness (mathematics)
• Functional analysis
• General topology