Compact Operator -- from Wolfram MathWorld
- ️Weisstein, Eric W.
- ️Mon Jul 29 2002
If 
and 
are Banach spaces and 
is a bounded linear operator, the 
is said to be a compact operator if it maps the unit
ball of 
into a relatively compact subset of 
(that is, a subset of 
with compact closure).
The basic example of a compact operator is an infinite diagonal matrix 
with 
.
The matrix gives a bounded map 
, where 
is the set of square-integrable sequences. It is a compact
operator because it is the limit of the finite rank matrices 
, which have the same entries as 
except 
for 
. That is, the 
have only finitely many nonzero entries.
The properties of compact operators are similar to those of finite-dimensional linear transformations. For Hilbert
spaces, any compact operator 
is the limit of a sequence of operators 
with finite rank, i.e., the image of 
is a finite-dimensional subspace in 
. However, this property does not hold in general as shown
by Enflo (1973), who constructed a Banach space that
provides a counterexample, thus solving the approximation
problem in the negative.
See also
Approximation Problem, Banach Space, Hilbert Space, Matrix, Nuclear Operator
Portions of this entry contributed by Todd Rowland
Portions of this entry contributed by José Carlos Santos
Explore with Wolfram|Alpha
References
Enflo, P. "A Counterexample to the Approximation Problem in Banach Spaces." Acta Math. 130, 309-317, 1973.
Referenced on Wolfram|Alpha
Cite this as:
Rowland, Todd; Santos, José Carlos; and Weisstein, Eric W. "Compact Operator." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompactOperator.html