<s>
In	O
mathematics	O
,	O
specifically	O
in	O
order	O
theory	O
and	O
functional	B-Application
analysis	I-Application
,	O
the	O
order	B-Algorithm
bound	I-Algorithm
dual	I-Algorithm
of	O
an	O
ordered	O
vector	O
space	O
is	O
the	O
set	O
of	O
all	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
that	O
map	O
order	O
intervals	O
,	O
which	O
are	O
sets	O
of	O
the	O
form	O
to	O
bounded	O
sets	O
.	O
</s>
<s>
The	O
order	B-Algorithm
bound	I-Algorithm
dual	I-Algorithm
of	O
is	O
denoted	O
by	O
This	O
space	O
plays	O
an	O
important	O
role	O
in	O
the	O
theory	O
of	O
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
The	O
positive	O
elements	O
of	O
the	O
order	B-Algorithm
bound	I-Algorithm
dual	I-Algorithm
form	O
a	O
cone	O
that	O
induces	O
an	O
ordering	O
on	O
called	O
the	O
.	O
</s>
<s>
If	O
is	O
an	O
ordered	O
vector	O
space	O
whose	O
positive	O
cone	O
is	O
generating	O
(	O
meaning	O
)	O
then	O
the	O
order	B-Algorithm
bound	I-Algorithm
dual	I-Algorithm
with	O
the	O
canonical	O
ordering	O
is	O
an	O
ordered	O
vector	O
space	O
.	O
</s>
<s>
The	O
order	B-Algorithm
bound	I-Algorithm
dual	I-Algorithm
of	O
an	O
ordered	O
vector	O
spaces	O
contains	O
its	O
order	B-Algorithm
dual	I-Algorithm
.	O
</s>
<s>
If	O
the	O
positive	O
cone	O
of	O
an	O
ordered	O
vector	O
space	O
is	O
generating	O
and	O
if	O
for	O
all	O
positive	O
and	O
we	O
have	O
then	O
the	O
order	B-Algorithm
dual	I-Algorithm
is	O
equal	O
to	O
the	O
order	B-Algorithm
bound	I-Algorithm
dual	I-Algorithm
,	O
which	O
is	O
an	O
order	O
complete	O
vector	B-Algorithm
lattice	I-Algorithm
under	O
its	O
canonical	O
ordering	O
.	O
</s>
