Let 
be a complex Hilbert space, and define a nest as
a set 
of closed subspaces of 
satisfying the conditions:
1. 
,
2. If 
,
then either 
or 
,
3. If 
,then
,
4. If 
,then the norm closure of the linear span of 
lies in 
.
(Davidson 1988).
The nest algebra associated with the nest 
is the set 
.
For example, consider an orthonormal basis 
of a separable Hilbert space 
. Put 
. Then 
is a nest and the associated nest
algebra 
is the algebra of operators whose matrix representation with respect to 
is upper triangular.
