A Hilbert-Schmidt operator 
on a Hilbert space 
with orthonormal basis 
for 
is an operator for which 
. The Hilbert-Schmidt operators
on 
form the set 
,
which is a self-adjoint ideal of 
.
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 
).
