Compact embedding explained
In 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. The notation for "
is compactly embedded in
" is
, or
.
When used in functional analysis, compact embedding is usually about 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.
Definition
Topological spaces
Let
be a
topological space, and let
and
be
subsets of
. We say that
is
compactly embedded in
if
V\subseteq\operatorname{Cl}(V)\subseteq\operatorname{Int}(W)
, where
denotes the
closure of
, and
denotes the
interior of
; and
is
compact.For
Hausdorff spaces, this is equivalent to there existing some compact set
such that
V\subseteqK\subseteq\operatorname{Int}(W)
.
Normed spaces
Let
and
be two
normed vector spaces with norms
and
respectively, and suppose that
. We say that
is
compactly embedded in
, if
is
continuously embedded in
; i.e., there is a constant
such that
for all
in
; and
into
is a
compact operator: any
bounded set in
is
totally bounded in
, i.e. every
sequence in such a bounded set has a
subsequence that is
Cauchy in the norm
.
Banach spaces
If
is a
Banach space, an equivalent definition is that the embedding operator (the identity)
is a
compact operator.
References
- Book: Adams . Robert A. . Sobolev Spaces . . Boston, MA . 978-0-12-044150-1 . 1975 .
- Book: Evans, Lawrence C. . Lawrence C. Evans . Partial differential equations . Providence, RI . American Mathematical Society . 1998 . 0-8218-0772-2.
- Book: Renardy, M. . Rogers, R. C. . amp . An Introduction to Partial Differential Equations . Berlin . Springer-Verlag . 1992 . 3-540-97952-2 .
