Compactly embedded
From Wikipedia, the free encyclopedia
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.
[edit] Definition (topological spaces)
Let 
be a topological space, and let V and W be subsets of X. We say that V is compactly embedded in W, and write
if
, where 
denotes the 
denotes the [edit] Definition (normed spaces)
Let X and Y be two normed vector spaces with norms 
and 
respectively, and suppose that 
. We say that X is compactly embedded in Y, and write
if
- X is continuously embedded in Y; i.e., there is a constant C such that
for all 
; and - 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 .
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.
[edit] References
- Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2.
- Rennardy, M., & Rogers, R.C. (1992). An Introduction to Partial Differential Equations. Berlin: Springer-Verlag. ISBN 3-540-97952-2.
