<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
branch	O
of	O
mathematics	O
,	O
nest	B-Algorithm
algebras	I-Algorithm
are	O
a	O
class	O
of	O
operator	B-Algorithm
algebras	I-Algorithm
that	O
generalise	O
the	O
upper-triangular	B-Algorithm
matrix	I-Algorithm
algebras	O
to	O
a	O
Hilbert	O
space	O
context	O
.	O
</s>
<s>
They	O
are	O
non-selfadjoint	O
algebras	O
,	O
are	O
closed	O
in	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
and	O
are	O
reflexive	O
.	O
</s>
<s>
Nest	B-Algorithm
algebras	I-Algorithm
are	O
among	O
the	O
simplest	O
examples	O
of	O
commutative	O
subspace	O
lattice	O
algebras	O
.	O
</s>
<s>
By	O
way	O
of	O
an	O
example	O
,	O
let	O
us	O
apply	O
this	O
definition	O
to	O
recover	O
the	O
finite-dimensional	O
upper-triangular	B-Algorithm
matrices	O
.	O
</s>
<s>
then	O
N	O
is	O
a	O
subspace	O
nest	O
,	O
and	O
the	O
corresponding	O
nest	B-Algorithm
algebra	I-Algorithm
of	O
nn	O
complex	O
matrices	O
M	O
leaving	O
each	O
subspace	O
in	O
N	O
invariant	O
that	O
is	O
,	O
satisfying	O
for	O
each	O
S	O
in	O
N	O
is	O
precisely	O
the	O
set	O
of	O
upper-triangular	B-Algorithm
matrices	O
.	O
</s>
<s>
If	O
we	O
omit	O
one	O
or	O
more	O
of	O
the	O
subspaces	O
Sj	O
from	O
N	O
then	O
the	O
corresponding	O
nest	B-Algorithm
algebra	I-Algorithm
consists	O
of	O
block	O
upper-triangular	B-Algorithm
matrices	O
.	O
</s>
<s>
Nest	B-Algorithm
algebras	I-Algorithm
are	O
hyperreflexive	O
with	O
distance	O
constant	O
1	O
.	O
</s>
