A compact operator 
between Banach spaces 
and 
is a bounded linear
operator that 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-summable sequences. It is a compact operator because it is the limit in the operator norm
of the finite rank matrices 
, which have the same
entries as 
except 
for 
.
That is, the 
have only finitely many nonzero
entries.
More generally, a diagonal bounded operator 
is compact iff 
. For the truncations above,
 |
where 
is the operator norm. Thus 
gives finite-rank
approximation in operator norm (Axler 2026, p. 315).
Conversely, if the diagonal does not tend to zero, some subsequence
satisfies 
.
The images of the corresponding standard basis vectors then have pairwise distances
at least 
,
so no subsequence converges.
This contradicts compactness. The original square-summability condition is therefore
sufficient but is not necessary.
The properties of compact operators are similar to those of finite-dimensional linear
transformations. For Hilbert spaces, any compact
operator 
is the limit in the operator
norm of a sequence of linear
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.
