<s>
Normal	B-Algorithm
cones	I-Algorithm
play	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
and	O
topological	B-Algorithm
vector	I-Algorithm
lattices	I-Algorithm
.	O
</s>
<s>
is	O
a	O
normal	B-Algorithm
cone	I-Algorithm
.	O
</s>
<s>
and	O
if	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
if	O
the	O
dual	O
cone	O
of	O
is	O
denoted	O
by	O
then	O
we	O
may	O
add	O
to	O
this	O
list	O
:	O
</s>
<s>
and	O
if	O
is	O
an	O
infrabarreled	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
if	O
is	O
the	O
family	O
of	O
all	O
strongly	O
bounded	O
subsets	O
of	O
then	O
we	O
may	O
add	O
to	O
this	O
list	O
:	O
</s>
<s>
and	O
if	O
is	O
an	O
ordered	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
over	O
the	O
reals	O
whose	O
positive	O
cone	O
is	O
then	O
we	O
may	O
add	O
to	O
this	O
list	O
:	O
</s>
<s>
<	O
li>there	O
exists	O
a	O
Hausdorff	O
locally	O
compact	O
topological	O
space	O
such	O
that	O
is	O
isomorphic	O
(	O
as	O
an	O
ordered	B-Algorithm
TVS	I-Algorithm
)	O
with	O
a	O
subspace	O
of	O
where	O
is	O
the	O
space	O
of	O
all	O
real-valued	O
continuous	O
functions	O
on	O
under	O
the	O
topology	O
of	O
compact	O
convergence	O
.	O
</s>
<s>
if	O
is	O
a	O
-cone	O
then	O
is	O
a	O
normal	B-Algorithm
cone	I-Algorithm
for	O
the	O
-topology	O
on	O
;	O
</s>
<s>
Suppose	O
is	O
an	O
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
,	O
and	O
we	O
define	O
whenever	O
lies	O
in	O
the	O
cone	O
.	O
</s>
<s>
If	O
is	O
a	O
Hausdorff	O
TVS	O
then	O
every	O
normal	B-Algorithm
cone	I-Algorithm
in	O
is	O
a	O
proper	O
cone	O
.	O
</s>
<s>
Suppose	O
that	O
the	O
positive	O
cone	O
of	O
an	O
ordered	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
weakly	O
normal	O
in	O
and	O
that	O
is	O
an	O
ordered	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
with	O
positive	O
cone	O
If	O
then	O
is	O
dense	O
in	O
where	O
is	O
the	O
canonical	O
positive	O
cone	O
of	O
and	O
is	O
the	O
space	O
with	O
the	O
topology	O
of	O
simple	O
convergence	O
.	O
</s>
<s>
If	O
the	O
topology	O
on	O
is	O
locally	B-Algorithm
convex	I-Algorithm
then	O
the	O
closure	O
of	O
a	O
normal	B-Algorithm
cone	I-Algorithm
is	O
a	O
normal	B-Algorithm
cone	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
the	O
closure	O
of	O
a	O
normal	B-Algorithm
cone	I-Algorithm
is	O
a	O
normal	B-Algorithm
cone	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
cone	O
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
and	O
if	O
is	O
the	O
dual	O
cone	O
of	O
then	O
if	O
and	O
only	O
if	O
is	O
weakly	O
normal	O
.	O
</s>
<s>
Every	O
normal	B-Algorithm
cone	I-Algorithm
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
weakly	O
normal	O
.	O
</s>
