<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
a	O
metrizable	O
(	O
resp	O
.	O
</s>
<s>
pseudometrizable	O
)	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
a	O
TVS	O
whose	O
topology	O
is	O
induced	O
by	O
a	O
metric	O
(	O
resp	O
.	O
</s>
<s>
An	O
LM-space	B-Algorithm
is	O
an	O
inductive	O
limit	O
of	O
a	O
sequence	O
of	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
A	O
pseudometric	O
on	O
is	O
called	O
a	O
ultrapseudometric	B-Algorithm
or	O
a	O
strong	O
pseudometric	O
if	O
it	O
satisfies	O
:	O
</s>
<s>
Strong/Ultrametric	O
triangle	O
inequality	O
:	O
</s>
<s>
ultrapseudometric	B-Algorithm
space	I-Algorithm
)	O
when	O
is	O
a	O
metric	O
(	O
resp	O
.	O
</s>
<s>
ultrapseudometric	B-Algorithm
)	O
.	O
</s>
<s>
A	O
topology	O
on	O
a	O
real	O
or	O
complex	O
vector	O
space	O
is	O
called	O
a	O
vector	O
topology	O
or	O
a	O
TVS	O
topology	O
if	O
it	O
makes	O
the	O
operations	O
of	O
vector	O
addition	O
and	O
scalar	O
multiplication	O
continuous	O
(	O
that	O
is	O
,	O
if	O
it	O
makes	O
into	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
)	O
.	O
</s>
<s>
Every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
an	O
additive	O
commutative	O
topological	O
group	O
but	O
not	O
all	O
group	O
topologies	O
on	O
are	O
vector	O
topologies	O
.	O
</s>
<s>
These	O
functions	O
can	O
then	O
be	O
used	O
to	O
prove	O
many	O
of	O
the	O
basic	O
properties	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
also	O
show	O
that	O
a	O
Hausdorff	O
TVS	O
with	O
a	O
countable	O
basis	O
of	O
neighborhoods	O
is	O
metrizable	O
.	O
</s>
<s>
If	O
and	O
are	O
paranorms	O
on	O
then	O
so	O
is	O
Moreover	O
,	O
and	O
This	O
makes	O
the	O
set	O
of	O
paranorms	O
on	O
into	O
a	O
conditionally	O
complete	B-Algorithm
lattice	O
.	O
</s>
<s>
Then	O
is	O
an	O
F-seminorm	O
on	O
that	O
induces	O
the	O
same	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
as	O
the	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
The	O
Fréchet	O
combination	O
can	O
be	O
generalized	O
by	O
use	O
of	O
a	O
bounded	B-Algorithm
remetrization	O
function	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
family	O
of	O
non-negative	O
F-seminorm	O
on	O
a	O
vector	O
space	O
is	O
a	O
bounded	B-Algorithm
remetrization	O
function	O
,	O
and	O
is	O
a	O
sequence	O
of	O
positive	O
real	O
numbers	O
whose	O
sum	O
is	O
finite	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
(	O
where	O
note	O
in	O
particular	O
that	O
is	O
assumed	O
to	O
be	O
a	O
vector	O
topology	O
)	O
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
is	O
locally	B-Algorithm
convex	I-Algorithm
and	O
pseudometrizable	O
.	O
</s>
<s>
If	O
is	O
a	O
complete	B-Algorithm
pseudometrizable	B-Algorithm
TVS	I-Algorithm
and	O
is	O
a	O
closed	O
vector	O
subspace	O
of	O
then	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
If	O
is	O
metrizable	B-Algorithm
TVS	I-Algorithm
and	O
is	O
a	O
closed	O
vector	O
subspace	O
of	O
then	O
is	O
metrizable	O
.	O
</s>
<s>
If	O
a	O
TVS	O
has	O
a	O
bounded	B-Algorithm
neighborhood	O
of	O
the	O
origin	O
then	O
it	O
is	O
pseudometrizable	O
;	O
the	O
converse	O
is	O
in	O
general	O
false	O
.	O
</s>
<s>
If	O
a	O
Hausdorff	O
TVS	O
has	O
a	O
bounded	B-Algorithm
neighborhood	O
of	O
the	O
origin	O
then	O
it	O
is	O
metrizable	O
.	O
</s>
<s>
Suppose	O
is	O
either	O
a	O
DF-space	B-Algorithm
or	O
an	O
LM-space	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
sequential	O
space	O
then	O
it	O
is	O
either	O
metrizable	O
or	O
else	O
a	O
Montel	B-Algorithm
DF-space	B-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
(	O
such	O
as	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
)	O
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
DF-space	B-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
reflexive	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
strong	O
bidual	O
(	O
that	O
is	O
,	O
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
)	O
of	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
its	O
strong	B-Algorithm
dual	I-Algorithm
has	O
one	O
of	O
the	O
following	O
properties	O
,	O
if	O
and	O
only	O
if	O
it	O
has	O
all	O
of	O
these	O
properties	O
:	O
(	O
1	O
)	O
bornological	B-Algorithm
,	O
(	O
2	O
)	O
infrabarreled	B-Algorithm
,	O
(	O
3	O
)	O
barreled	B-Algorithm
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
seminormable	O
if	O
and	O
only	O
if	O
it	O
has	O
a	O
convex	O
bounded	B-Algorithm
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Every	O
metrizable	B-Algorithm
TVS	I-Algorithm
on	O
a	O
finite-dimensional	O
vector	O
space	O
is	O
a	O
normable	O
locally	B-Algorithm
convex	I-Algorithm
complete	B-Algorithm
TVS	I-Algorithm
,	O
being	O
TVS-isomorphic	O
to	O
Euclidean	O
space	O
.	O
</s>
<s>
Consequently	O
,	O
any	O
metrizable	B-Algorithm
TVS	I-Algorithm
that	O
is	O
normable	O
must	O
be	O
infinite	O
dimensional	O
.	O
</s>
<s>
If	O
is	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
TVS	I-Algorithm
that	O
possess	O
a	O
countable	O
fundamental	O
system	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
,	O
then	O
is	O
normable	O
.	O
</s>
<s>
If	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
has	O
a	O
(	O
von	O
Neumann	O
)	O
bounded	B-Algorithm
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
is	O
normable	O
.	O
</s>
<s>
and	O
if	O
this	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
also	O
metrizable	O
,	O
then	O
the	O
following	O
may	O
be	O
appended	O
to	O
this	O
list	O
:	O
</s>
<s>
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
is	O
metrizable	O
.	O
</s>
<s>
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
is	O
a	O
Fréchet	O
–	O
Urysohn	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
(	O
such	O
as	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
)	O
is	O
normable	O
then	O
its	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
is	O
not	O
a	O
Fréchet	O
–	O
Urysohn	O
space	O
and	O
consequently	O
,	O
this	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
also	O
neither	O
metrizable	O
nor	O
normable	O
.	O
</s>
<s>
Another	O
consequence	O
of	O
this	O
is	O
that	O
if	O
is	O
a	O
reflexive	O
locally	B-Algorithm
convex	I-Algorithm
TVS	I-Algorithm
whose	O
strong	B-Algorithm
dual	I-Algorithm
is	O
metrizable	O
then	O
is	O
necessarily	O
a	O
reflexive	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
is	O
a	O
DF-space	B-Algorithm
,	O
both	O
and	O
are	O
necessarily	O
complete	B-Algorithm
Hausdorff	O
ultrabornological	B-Algorithm
distinguished	B-Algorithm
webbed	B-Algorithm
spaces	I-Algorithm
,	O
and	O
moreover	O
,	O
is	O
normable	O
if	O
and	O
only	O
if	O
is	O
normable	O
if	O
and	O
only	O
if	O
is	O
Fréchet	O
–	O
Urysohn	O
if	O
and	O
only	O
if	O
is	O
metrizable	O
.	O
</s>
<s>
The	O
set	O
is	O
metrically	O
bounded	B-Algorithm
or	O
-bounded	O
if	O
there	O
exists	O
a	O
real	O
number	O
such	O
that	O
for	O
all	O
;	O
</s>
<s>
If	O
is	O
bounded	B-Algorithm
in	O
a	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
then	O
it	O
is	O
metrically	O
bounded	B-Algorithm
;	O
</s>
<s>
the	O
converse	O
is	O
in	O
general	O
false	O
but	O
it	O
is	O
true	O
for	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	O
TVSs	O
.	O
</s>
<s>
Every	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
TVS	I-Algorithm
is	O
a	O
quasibarrelled	B-Algorithm
space	I-Algorithm
,	O
bornological	B-Algorithm
space	I-Algorithm
,	O
and	O
a	O
Mackey	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
complete	B-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
and	O
a	O
Baire	O
space	O
(	O
and	O
hence	O
non-meager	O
)	O
.	O
</s>
<s>
However	O
,	O
there	O
exist	O
metrizable	O
Baire	O
spaces	O
that	O
are	O
not	O
complete	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
then	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
is	O
bornological	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
barreled	B-Algorithm
,	O
if	O
and	O
only	O
if	O
it	O
is	O
infrabarreled	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
complete	B-Algorithm
pseudometrizable	B-Algorithm
TVS	I-Algorithm
and	O
is	O
a	O
closed	O
vector	O
subspace	O
of	O
then	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
a	O
webbed	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
complete	B-Algorithm
metrizable	O
TVSs	O
(	O
i.e.	O
</s>
<s>
F-spaces	B-Algorithm
)	O
and	O
if	O
is	O
coarser	O
than	O
then	O
;	O
this	O
is	O
no	O
longer	O
guaranteed	O
to	O
be	O
true	O
if	O
any	O
one	O
of	O
these	O
metrizable	O
TVSs	O
is	O
not	O
complete	B-Algorithm
.	O
</s>
<s>
Said	O
differently	O
,	O
if	O
and	O
are	O
both	O
F-spaces	B-Algorithm
but	O
with	O
different	O
topologies	O
,	O
then	O
neither	O
one	O
of	O
and	O
contains	O
the	O
other	O
as	O
a	O
subset	O
.	O
</s>
<s>
also	O
be	O
complete	B-Algorithm
)	O
is	O
if	O
these	O
two	O
norms	O
and	O
are	O
equivalent	O
;	O
if	O
they	O
are	O
not	O
equivalent	O
,	O
then	O
can	O
not	O
be	O
a	O
Banach	O
space	O
.	O
</s>
<s>
As	O
another	O
consequence	O
,	O
if	O
is	O
a	O
Banach	O
space	O
and	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
then	O
the	O
map	O
is	O
continuous	O
if	O
and	O
only	O
if	O
the	O
Fréchet	B-Algorithm
space	I-Algorithm
the	O
TVS	O
(	O
here	O
,	O
the	O
Banach	O
space	O
is	O
being	O
considered	O
as	O
a	O
TVS	O
,	O
which	O
means	O
that	O
its	O
norm	O
is	O
"	O
forgetten	O
"	O
but	O
its	O
topology	O
is	O
remembered	O
)	O
.	O
</s>
<s>
A	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
normable	O
if	O
and	O
only	O
if	O
its	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
is	O
a	O
Fréchet	O
–	O
Urysohn	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Any	O
product	O
of	O
complete	B-Algorithm
metrizable	O
TVSs	O
is	O
a	O
Baire	O
space	O
.	O
</s>
<s>
Every	O
complete	B-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
and	O
a	O
Baire	O
space	O
(	O
and	O
thus	O
non-meager	O
)	O
.	O
</s>
<s>
The	O
dimension	O
of	O
a	O
complete	B-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
either	O
finite	O
or	O
uncountable	O
.	O
</s>
<s>
Every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
and	O
more	O
generally	O
,	O
a	O
topological	O
group	O
)	O
has	O
a	O
canonical	O
uniform	O
structure	O
,	O
induced	O
by	O
its	O
topology	O
,	O
which	O
allows	O
the	O
notions	O
of	O
completeness	O
and	O
uniform	O
continuity	O
to	O
be	O
applied	O
to	O
it	O
.	O
</s>
<s>
If	O
is	O
a	O
metrizable	B-Algorithm
TVS	I-Algorithm
and	O
is	O
a	O
metric	O
that	O
defines	O
'	O
s	O
topology	O
,	O
then	O
its	O
possible	O
that	O
is	O
complete	B-Algorithm
as	O
a	O
TVS	O
(	O
i.e.	O
</s>
<s>
relative	O
to	O
its	O
uniformity	O
)	O
but	O
the	O
metric	O
is	O
a	O
complete	B-Algorithm
metric	O
(	O
such	O
metrics	O
exist	O
even	O
for	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
closed	O
vector	O
subspace	O
of	O
a	O
complete	B-Algorithm
pseudometrizable	B-Algorithm
TVS	I-Algorithm
then	O
the	O
quotient	O
space	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
separable	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
and	O
let	O
be	O
its	O
completion	O
.	O
</s>
<s>
In	O
a	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
,	O
every	O
bornivore	B-Algorithm
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
subset	O
of	O
a	O
complete	B-Algorithm
metric	O
space	O
is	O
closed	O
if	O
and	O
only	O
if	O
it	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
If	O
a	O
space	O
is	O
not	O
complete	B-Algorithm
,	O
then	O
is	O
a	O
closed	O
subset	O
of	O
that	O
is	O
not	O
complete	B-Algorithm
.	O
</s>
<s>
If	O
a	O
generalized	O
series	O
converges	O
in	O
a	O
metrizable	B-Algorithm
TVS	I-Algorithm
,	O
then	O
the	O
set	O
is	O
necessarily	O
countable	O
(	O
that	O
is	O
,	O
either	O
finite	O
or	O
countably	O
infinite	O
)	O
;	O
</s>
<s>
If	O
is	O
a	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
and	O
maps	O
bounded	B-Algorithm
subsets	O
of	O
to	O
bounded	B-Algorithm
subsets	O
of	O
then	O
is	O
continuous	O
.	O
</s>
<s>
Discontinuous	O
linear	O
functionals	O
exist	O
on	O
any	O
infinite-dimensional	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
a	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
is	O
finite-dimensional	O
if	O
and	O
only	O
if	O
its	O
continuous	O
dual	O
space	O
is	O
equal	O
to	O
its	O
algebraic	O
dual	O
space	O
.	O
</s>
<s>
is	O
a	O
(	O
locally	O
)	O
bounded	B-Algorithm
map	O
(	O
that	O
is	O
,	O
maps	O
(	O
von	O
Neumann	O
)	O
bounded	B-Algorithm
subsets	O
of	O
to	O
bounded	B-Algorithm
subsets	O
of	O
)	O
;	O
</s>
<s>
the	O
image	O
under	O
of	O
every	O
null	O
sequence	O
in	O
is	O
a	O
bounded	B-Algorithm
set	I-Algorithm
where	O
by	O
definition	O
,	O
a	O
is	O
a	O
sequence	O
that	O
converges	O
to	O
the	O
origin	O
.	O
</s>
<s>
Theorem	O
:	O
If	O
is	O
a	O
complete	B-Algorithm
pseudometrizable	B-Algorithm
TVS	I-Algorithm
,	O
is	O
a	O
Hausdorff	O
TVS	O
,	O
and	O
is	O
a	O
closed	O
and	O
almost	B-Algorithm
open	I-Algorithm
linear	O
surjection	O
,	O
then	O
is	O
an	O
open	O
map	O
.	O
</s>
<s>
Theorem	O
:	O
If	O
is	O
a	O
surjective	O
linear	O
operator	O
from	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
onto	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
(	O
e.g.	O
</s>
<s>
every	O
complete	B-Algorithm
pseudometrizable	O
space	O
is	O
barrelled	O
)	O
then	O
is	O
almost	B-Algorithm
open	I-Algorithm
.	O
</s>
<s>
Theorem	O
:	O
If	O
is	O
a	O
surjective	O
linear	O
operator	O
from	O
a	O
TVS	O
onto	O
a	O
Baire	O
space	O
then	O
is	O
almost	B-Algorithm
open	I-Algorithm
.	O
</s>
<s>
Theorem	O
:	O
Suppose	O
is	O
a	O
continuous	O
linear	O
operator	O
from	O
a	O
complete	B-Algorithm
pseudometrizable	B-Algorithm
TVS	I-Algorithm
into	O
a	O
Hausdorff	O
TVS	O
If	O
the	O
image	O
of	O
is	O
non-meager	O
in	O
then	O
is	O
a	O
surjective	O
open	O
map	O
and	O
is	O
a	O
complete	B-Algorithm
metrizable	O
space	O
.	O
</s>
<s>
The	O
Hahn-Banach	O
theorem	O
guarantees	O
that	O
every	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
has	O
the	O
HBEP	O
.	O
</s>
<s>
For	O
complete	B-Algorithm
metrizable	O
TVSs	O
there	O
is	O
a	O
converse	O
:	O
</s>
<s>
If	O
a	O
vector	O
space	O
has	O
uncountable	O
dimension	O
and	O
if	O
we	O
endow	O
it	O
with	O
the	O
finest	O
vector	O
topology	O
then	O
this	O
is	O
a	O
TVS	O
with	O
the	O
HBEP	O
that	O
is	O
neither	O
locally	B-Algorithm
convex	I-Algorithm
or	O
metrizable	O
.	O
</s>
