<s>
The	O
order	B-Algorithm
dual	I-Algorithm
of	O
is	O
denoted	O
by	O
.	O
</s>
<s>
Along	O
with	O
the	O
related	O
concept	O
of	O
the	O
order	B-Algorithm
bound	I-Algorithm
dual	I-Algorithm
,	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
dual	I-Algorithm
form	O
a	O
cone	O
that	O
induces	O
an	O
ordering	O
on	O
called	O
the	O
canonical	O
ordering	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
that	O
is	O
,	O
)	O
then	O
the	O
order	B-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
dual	I-Algorithm
is	O
the	O
span	O
of	O
the	O
set	O
of	O
positive	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
.	O
</s>
<s>
The	O
order	B-Algorithm
dual	I-Algorithm
is	O
contained	O
in	O
the	O
order	B-Algorithm
bound	I-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
holds	O
for	O
all	O
positive	O
and	O
,	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	B-Algorithm
complete	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
under	O
its	O
canonical	O
ordering	O
.	O
</s>
<s>
The	O
order	B-Algorithm
dual	I-Algorithm
of	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
is	O
an	O
order	B-Algorithm
complete	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
.	O
</s>
<s>
The	O
order	B-Algorithm
dual	I-Algorithm
of	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
can	O
be	O
finite	O
dimension	O
(	O
possibly	O
even	O
)	O
even	O
if	O
is	O
infinite-dimensional	O
.	O
</s>
<s>
Then	O
the	O
order	B-Algorithm
bidual	I-Algorithm
is	O
defined	O
to	O
be	O
the	O
order	B-Algorithm
dual	I-Algorithm
of	O
and	O
is	O
denoted	O
by	O
.	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
holds	O
for	O
all	O
positive	O
and	O
,	O
then	O
is	O
an	O
order	B-Algorithm
complete	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
and	O
the	O
evaluation	O
map	O
is	O
order	O
preserving	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
then	O
is	O
an	O
order	B-Algorithm
complete	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
and	O
if	O
is	O
a	O
solid	B-Algorithm
subspace	I-Algorithm
of	O
that	O
separates	O
points	O
in	O
,	O
then	O
the	O
evaluation	O
map	O
defined	O
by	O
sending	O
to	O
the	O
map	O
given	O
by	O
,	O
is	O
a	O
lattice	O
isomorphism	O
of	O
onto	O
a	O
vector	O
sublattice	O
of	O
.	O
</s>
<s>
However	O
,	O
the	O
image	O
of	O
this	O
map	O
is	O
in	O
general	O
not	O
order	B-Algorithm
complete	I-Algorithm
even	O
if	O
is	O
order	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
a	O
regularly	O
ordered	O
,	O
order	B-Algorithm
complete	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
need	O
not	O
be	O
mapped	O
by	O
the	O
evaluation	O
map	O
onto	O
a	O
band	O
in	O
the	O
order	B-Algorithm
bidual	I-Algorithm
.	O
</s>
<s>
An	O
order	B-Algorithm
complete	I-Algorithm
,	O
regularly	O
ordered	O
vector	B-Algorithm
lattice	I-Algorithm
whose	O
canonical	O
image	O
in	O
its	O
order	B-Algorithm
bidual	I-Algorithm
is	O
order	B-Algorithm
complete	I-Algorithm
is	O
called	O
minimal	O
and	O
is	O
said	O
to	O
be	O
of	O
minimal	O
type	O
.	O
</s>
<s>
For	O
any	O
,	O
the	O
Banach	O
lattice	O
is	O
order	B-Algorithm
complete	I-Algorithm
and	O
of	O
minimal	O
type	O
;	O
</s>
<s>
in	O
particular	O
,	O
the	O
norm	O
topology	O
on	O
this	O
space	O
is	O
the	O
finest	O
locally	O
convex	O
topology	O
for	O
which	O
every	O
order	B-Algorithm
convergent	I-Algorithm
filter	O
converges	O
.	O
</s>
<s>
Let	O
be	O
an	O
order	B-Algorithm
complete	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
of	O
minimal	O
type	O
.	O
</s>
<s>
is	O
a	O
weak	B-Algorithm
order	I-Algorithm
unit	I-Algorithm
.	O
</s>
<s>
For	O
every	O
non-0	O
positive	O
linear	B-Algorithm
functional	I-Algorithm
on	O
,	O
</s>
<s>
For	O
each	O
topology	O
on	O
such	O
that	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
,	O
is	O
a	O
quasi-interior	B-Algorithm
point	I-Algorithm
of	O
its	O
positive	O
cone	O
.	O
</s>
<s>
An	O
ordered	O
vector	O
space	O
is	O
called	O
regularly	O
ordered	O
and	O
its	O
order	O
is	O
said	O
to	O
be	O
regular	O
if	O
it	O
is	O
Archimedean	B-Algorithm
ordered	I-Algorithm
and	O
distinguishes	O
points	O
in	O
.	O
</s>
