<s>
In	O
other	O
words	O
,	O
a	O
positive	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
maps	O
the	O
positive	O
cone	O
of	O
the	O
domain	B-Algorithm
into	O
the	O
positive	O
cone	O
of	O
the	O
codomain	B-Algorithm
.	O
</s>
<s>
Every	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
is	O
a	O
type	O
of	O
positive	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
The	O
significance	O
of	O
positive	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
lies	O
in	O
results	O
such	O
as	O
Riesz	O
–	O
Markov	O
–	O
Kakutani	O
representation	O
theorem	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
positive	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
in	O
is	O
a	O
cone	O
in	O
that	O
defines	O
a	O
preorder	O
on	O
.	O
</s>
<s>
If	O
and	O
are	O
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
and	O
if	O
is	O
a	O
family	O
of	O
bounded	O
subsets	O
of	O
whose	O
union	O
covers	O
then	O
the	O
positive	O
cone	O
in	O
,	O
which	O
is	O
the	O
space	O
of	O
all	O
continuous	O
linear	B-Architecture
maps	I-Architecture
from	O
into	O
is	O
closed	O
in	O
when	O
is	O
endowed	O
with	O
the	O
-topology	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
of	O
dimension	O
greater	O
than	O
0	O
then	O
this	O
condition	O
is	O
also	O
necessary	O
.	O
</s>
<s>
Thus	O
,	O
if	O
the	O
positive	O
cone	O
of	O
is	O
total	O
in	O
and	O
if	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
then	O
the	O
canonical	O
ordering	O
of	O
defined	O
by	O
is	O
a	O
regular	O
order	O
.	O
</s>
<s>
Proposition	O
:	O
Suppose	O
that	O
and	O
are	O
ordered	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
with	O
being	O
a	O
Mackey	B-Algorithm
space	I-Algorithm
on	O
which	O
every	O
positive	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
is	O
continuous	O
.	O
</s>
<s>
If	O
the	O
positive	O
cone	O
of	O
is	O
a	O
weakly	B-Algorithm
normal	I-Algorithm
cone	I-Algorithm
in	O
then	O
every	O
positive	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
from	O
into	O
is	O
continuous	O
.	O
</s>
<s>
Proposition	O
:	O
Suppose	O
is	O
a	O
barreled	B-Algorithm
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	O
)	O
with	O
positive	O
cone	O
that	O
satisfies	O
and	O
is	O
a	O
semi-reflexive	O
ordered	B-Algorithm
TVS	I-Algorithm
with	O
a	O
positive	O
cone	O
that	O
is	O
a	O
normal	B-Algorithm
cone	I-Algorithm
.	O
</s>
