<s>
In	O
mathematics	O
,	O
specifically	O
in	O
functional	B-Application
analysis	I-Application
and	O
order	O
theory	O
,	O
a	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
that	O
has	O
a	O
partial	O
order	O
making	O
it	O
into	O
vector	B-Algorithm
lattice	I-Algorithm
that	O
is	O
possesses	O
a	O
neighborhood	O
base	O
at	O
the	O
origin	O
consisting	O
of	O
solid	B-Algorithm
sets	I-Algorithm
.	O
</s>
<s>
Ordered	O
vector	B-Algorithm
lattices	I-Algorithm
have	O
important	O
applications	O
in	O
spectral	O
theory	O
.	O
</s>
<s>
If	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
then	O
by	O
the	O
vector	B-Algorithm
lattice	I-Algorithm
operations	O
we	O
mean	O
the	O
following	O
maps	O
:	O
</s>
<s>
If	O
is	O
a	O
TVS	O
over	O
the	O
reals	O
and	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
,	O
then	O
is	O
locally	B-Algorithm
solid	I-Algorithm
if	O
and	O
only	O
if	O
(	O
1	O
)	O
its	O
positive	O
cone	O
is	O
a	O
normal	B-Algorithm
cone	I-Algorithm
,	O
and	O
(	O
2	O
)	O
the	O
vector	B-Algorithm
lattice	I-Algorithm
operations	O
are	O
continuous	O
.	O
</s>
<s>
If	O
is	O
a	O
vector	B-Algorithm
lattice	I-Algorithm
and	O
an	O
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
that	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
in	O
which	O
the	O
positive	O
cone	O
is	O
a	O
normal	B-Algorithm
cone	I-Algorithm
,	O
then	O
the	O
lattice	O
operations	O
are	O
continuous	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
and	O
an	O
ordered	O
vector	O
space	O
then	O
is	O
called	O
locally	B-Algorithm
solid	I-Algorithm
if	O
possesses	O
a	O
neighborhood	O
base	O
at	O
the	O
origin	O
consisting	O
of	O
solid	B-Algorithm
sets	I-Algorithm
.	O
</s>
<s>
A	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
a	O
Hausdorff	O
TVS	O
that	O
has	O
a	O
partial	O
order	O
making	O
it	O
into	O
vector	B-Algorithm
lattice	I-Algorithm
that	O
is	O
locally	B-Algorithm
solid	I-Algorithm
.	O
</s>
<s>
Every	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
has	O
a	O
closed	O
positive	O
cone	O
and	O
is	O
thus	O
an	O
ordered	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Let	O
denote	O
the	O
set	O
of	O
all	O
bounded	O
subsets	O
of	O
a	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
with	O
positive	O
cone	O
and	O
for	O
any	O
subset	O
,	O
let	O
be	O
the	O
-saturated	O
hull	O
of	O
.	O
</s>
<s>
Then	O
the	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
's	O
positive	O
cone	O
is	O
a	O
strict	O
-cone	O
,	O
where	O
is	O
a	O
strict	O
-cone	O
means	O
that	O
is	O
a	O
fundamental	O
subfamily	O
of	O
that	O
is	O
,	O
every	O
is	O
contained	O
as	O
a	O
subset	O
of	O
some	O
element	O
of	O
)	O
.	O
</s>
<s>
If	O
a	O
topological	B-Algorithm
vector	I-Algorithm
lattice	I-Algorithm
is	O
order	B-Algorithm
complete	I-Algorithm
then	O
every	O
band	O
is	O
closed	O
in	O
.	O
</s>
<s>
These	O
spaces	O
are	O
order	B-Algorithm
complete	I-Algorithm
for	O
.	O
</s>
