Let 
be a Hilbert space and 
is an orthonormal basis for 
. The set 
of all operators 
for which 
is a self-adjoint ideal of 
.
These operators are called Hilbert-Schmidt operators on 
.
The algebra 
with the Hilbert-Schmidt norm 
is a Banach
algebra. It contains operators of finite rank as a dense subset and is contained
in the space 
of compact operators. For any pair of operators 
and 
in 
, the family 
is summable. Its sum 
defines an inner product in 
and 
. So 
can be regarded as a Hilbert space (independent on the
choice basis 
).
