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.
