<s>
In	O
mathematics	O
,	O
specifically	O
in	O
order	O
theory	O
and	O
functional	B-Application
analysis	I-Application
,	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
(	O
LCVL	O
)	O
is	O
a	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
that	O
is	O
also	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
LCVLs	O
are	O
important	O
in	O
the	O
theory	O
of	O
topological	B-Algorithm
vector	I-Algorithm
lattices	I-Algorithm
.	O
</s>
<s>
The	O
Minkowski	O
functional	O
of	O
a	O
convex	O
,	O
absorbing	B-Algorithm
,	O
and	O
solid	B-Algorithm
set	I-Algorithm
is	O
a	O
called	O
a	O
lattice	B-Algorithm
semi-norm	I-Algorithm
.	O
</s>
<s>
The	O
topology	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
generated	O
by	O
the	O
family	O
of	O
all	O
continuous	O
lattice	B-Algorithm
semi-norms	I-Algorithm
.	O
</s>
<s>
Every	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
possesses	O
a	O
neighborhood	O
base	O
at	O
the	O
origin	O
consisting	O
of	O
convex	O
balanced	O
solid	B-Algorithm
absorbing	B-Algorithm
sets	I-Algorithm
.	O
</s>
<s>
The	O
strong	O
dual	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
an	O
order	B-Algorithm
complete	I-Algorithm
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
(	O
under	O
its	O
canonical	O
order	O
)	O
and	O
it	O
is	O
a	O
solid	B-Algorithm
subspace	I-Algorithm
of	O
the	O
order	B-Algorithm
dual	I-Algorithm
of	O
;	O
</s>
<s>
moreover	O
,	O
if	O
is	O
a	O
barreled	B-Algorithm
space	I-Algorithm
then	O
the	O
continuous	O
dual	O
space	O
of	O
is	O
a	O
band	O
in	O
the	O
order	B-Algorithm
dual	I-Algorithm
of	O
and	O
the	O
strong	O
dual	O
of	O
is	O
a	O
complete	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
.	O
</s>
<s>
If	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
barreled	B-Algorithm
then	O
its	O
strong	O
dual	O
space	O
is	O
complete	O
(	O
this	O
is	O
not	O
necessarily	O
true	O
if	O
the	O
space	O
is	O
merely	O
a	O
locally	B-Algorithm
convex	I-Algorithm
barreled	B-Algorithm
space	I-Algorithm
but	O
not	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
)	O
.	O
</s>
<s>
If	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
semi-reflexive	O
then	O
it	O
is	O
order	B-Algorithm
complete	I-Algorithm
and	O
(	O
that	O
is	O
,	O
)	O
is	O
a	O
complete	O
TVS	O
;	O
</s>
<s>
moreover	O
,	O
if	O
in	O
addition	O
every	O
positive	O
linear	O
functional	O
on	O
is	O
continuous	O
then	O
is	O
of	O
is	O
of	O
minimal	B-Algorithm
type	I-Algorithm
,	O
the	O
order	B-Algorithm
topology	I-Algorithm
on	O
is	O
equal	O
to	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
and	O
is	O
reflexive	O
.	O
</s>
<s>
Every	O
reflexive	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
order	B-Algorithm
complete	I-Algorithm
and	O
a	O
complete	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
whose	O
strong	O
dual	O
is	O
a	O
barreled	B-Algorithm
reflexive	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
that	O
can	O
be	O
identified	O
under	O
the	O
canonical	O
evaluation	O
map	O
with	O
the	O
strong	O
bidual	O
(	O
that	O
is	O
,	O
the	O
strong	O
dual	O
of	O
the	O
strong	O
dual	O
)	O
.	O
</s>
<s>
If	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
an	O
infrabarreled	B-Algorithm
TVS	O
then	O
it	O
can	O
be	O
identified	O
under	O
the	O
evaluation	O
map	O
with	O
a	O
topological	O
vector	O
sublattice	O
of	O
its	O
strong	O
bidual	O
,	O
which	O
is	O
an	O
order	B-Algorithm
complete	I-Algorithm
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
under	O
its	O
canonical	O
order	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
that	O
is	O
bornological	B-Algorithm
and	O
sequentially	B-Algorithm
complete	I-Algorithm
,	O
then	O
there	O
exists	O
a	O
family	O
of	O
compact	O
spaces	O
and	O
a	O
family	O
of	O
-indexed	O
vector	O
lattice	O
embeddings	O
such	O
that	O
is	O
the	O
finest	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
making	O
each	O
continuous	O
.	O
</s>
<s>
Every	O
Banach	O
lattice	O
,	O
normed	B-Algorithm
lattice	I-Algorithm
,	O
and	O
Fréchet	B-Algorithm
lattice	I-Algorithm
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
.	O
</s>
