Compact embedding

In mathematics, the notion of being compactly embedded expresses the idea that one set is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis.

Definition (topological spaces)

Let $(X,{\mathcal {T}})$ be a topological space, and let $V$ and $W$ be subsets of $X$. We say that $V$ is compactly embedded in $W$, and write

$V\subset \subset W,$

if $V\subseteq {\bar {V}}\subseteq \mathrm {Int} (W)$ and ${\bar {V}}$ is compact. ${\bar {V}}$ denotes the closure of $V$, and $\mathrm {Int} (W)$ denotes the interior of $W$, defined as $W\setminus \partial W$, i.e. $W$ without its boundary.

Definition (normed spaces)

Let $X$ and $Y$ be two normed vector spaces with norms $\|-\|_{X}$ and $\|-\|_{Y}$ respectively, and suppose that $X\subseteq Y$. We say that $X$ is compactly embedded in $Y$, and write

$X\subset \subset Y,$

• there is a constant $C$ such that $\|x\|_{Y}\leq C\|x\|_{Y}$ for all $x\in X$;
• any bounded set in $X$ is precompact in $Y$, i.e. every sequence in such a bounded set has a subsequence that is Cauchy in the norm $\|-\|_{Y}$.

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.