<s>
The	O
order	B-Algorithm
topology	I-Algorithm
is	O
an	O
important	O
topology	B-Architecture
that	O
is	O
used	O
frequently	O
in	O
the	O
theory	O
of	O
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
because	O
the	O
topology	B-Architecture
stems	O
directly	O
from	O
the	O
algebraic	O
and	O
order	O
theoretic	O
properties	O
of	O
rather	O
than	O
from	O
some	O
topology	B-Architecture
that	O
starts	O
out	O
having	O
.	O
</s>
<s>
For	O
many	O
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
that	O
occur	O
in	O
analysis	O
,	O
their	O
topologies	B-Architecture
are	O
identical	O
to	O
the	O
order	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
The	O
family	O
of	O
all	O
locally	B-Algorithm
convex	I-Algorithm
topologies	I-Algorithm
on	O
for	O
which	O
every	O
order	O
interval	O
is	O
bounded	O
is	O
non-empty	O
(	O
since	O
it	O
contains	O
the	O
coarsest	O
possible	O
topology	B-Architecture
on	O
)	O
and	O
the	O
order	B-Algorithm
topology	I-Algorithm
is	O
the	O
upper	O
bound	O
of	O
this	O
family	O
.	O
</s>
<s>
For	O
every	O
let	O
and	O
endow	O
with	O
its	O
order	B-Algorithm
topology	I-Algorithm
(	O
which	O
makes	O
it	O
into	O
a	O
normable	O
space	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
regularly	B-Algorithm
ordered	I-Algorithm
vector	O
space	O
over	O
the	O
reals	O
and	O
if	O
is	O
any	O
subset	O
of	O
the	O
positive	O
cone	O
of	O
that	O
is	O
cofinal	O
in	O
(	O
e.g.	O
</s>
<s>
could	O
be	O
)	O
,	O
then	O
with	O
its	O
order	B-Algorithm
topology	I-Algorithm
is	O
the	O
inductive	O
limit	O
of	O
(	O
where	O
the	O
bonding	O
maps	O
are	O
the	O
natural	O
inclusions	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
regularly	B-Algorithm
ordered	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
then	O
the	O
ordered	O
topology	B-Architecture
is	O
the	O
finest	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
topology	B-Architecture
on	O
making	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
.	O
</s>
<s>
If	O
in	O
addition	O
is	O
order	O
complete	O
then	O
with	O
the	O
order	B-Algorithm
topology	I-Algorithm
is	O
a	O
barreled	B-Algorithm
space	I-Algorithm
and	O
every	O
band	O
decomposition	O
of	O
is	O
a	O
topological	O
direct	O
sum	O
for	O
this	O
topology	B-Architecture
.	O
</s>
<s>
If	O
separates	O
points	O
in	O
(	O
such	O
as	O
if	O
is	O
regular	O
)	O
then	O
is	O
a	O
bornological	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
TVS	O
.	O
</s>
<s>
Each	O
positive	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
between	O
two	O
ordered	O
vector	O
spaces	O
is	O
continuous	O
for	O
the	O
respective	O
order	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
Each	O
order	O
unit	O
of	O
an	O
ordered	B-Algorithm
TVS	I-Algorithm
is	O
interior	O
to	O
the	O
positive	O
cone	O
for	O
the	O
order	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
If	O
the	O
order	O
of	O
an	O
ordered	O
vector	O
space	O
is	O
a	O
regular	O
order	O
and	O
if	O
each	O
positive	O
sequence	O
of	O
type	O
in	O
is	O
order	B-Algorithm
summable	I-Algorithm
,	O
then	O
endowed	O
with	O
its	O
order	B-Algorithm
topology	I-Algorithm
is	O
a	O
barreled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
the	O
order	O
of	O
an	O
ordered	O
vector	O
space	O
is	O
a	O
regular	O
order	O
and	O
if	O
for	O
all	O
and	O
holds	O
,	O
then	O
the	O
positive	O
cone	O
of	O
is	O
a	O
normal	B-Algorithm
cone	I-Algorithm
in	O
when	O
is	O
endowed	O
with	O
the	O
order	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
continuous	O
dual	O
space	O
of	O
with	O
the	O
order	B-Algorithm
topology	I-Algorithm
will	O
be	O
the	O
order	B-Algorithm
dual	I-Algorithm
+	O
.	O
</s>
<s>
If	O
is	O
an	O
Archimedean	B-Algorithm
ordered	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
over	O
the	O
real	O
numbers	O
having	O
an	O
order	O
unit	O
and	O
let	O
denote	O
the	O
order	B-Algorithm
topology	I-Algorithm
on	O
Then	O
is	O
an	O
ordered	B-Algorithm
TVS	I-Algorithm
that	O
is	O
normable	O
,	O
is	O
the	O
finest	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
topology	B-Architecture
on	O
such	O
that	O
the	O
positive	O
cone	O
is	O
normal	O
,	O
and	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
Each	O
positive	O
sequence	O
of	O
type	O
in	O
is	O
order	B-Algorithm
summable	I-Algorithm
.	O
</s>
<s>
A	O
vector	B-Algorithm
lattice	I-Algorithm
homomorphism	O
from	O
into	O
is	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
when	O
and	O
are	O
given	O
their	O
respective	O
order	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
The	O
order	B-Algorithm
topology	I-Algorithm
of	O
a	O
finite	O
product	O
of	O
ordered	O
vector	O
spaces	O
(	O
this	O
product	O
having	O
its	O
canonical	O
order	O
)	O
is	O
identical	O
to	O
the	O
product	O
topology	B-Architecture
of	O
the	O
topological	O
product	O
of	O
the	O
constituent	O
ordered	O
vector	O
spaces	O
(	O
when	O
each	O
is	O
given	O
its	O
order	B-Algorithm
topology	I-Algorithm
)	O
.	O
</s>
