<s>
The	O
metric	O
space	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
Thus	O
,	O
a	O
real	O
or	O
complex	O
F-space	B-Algorithm
is	O
equivalently	O
a	O
real	O
or	O
complex	O
vector	O
space	O
equipped	O
with	O
a	O
complete	B-Algorithm
F-norm	O
.	O
</s>
<s>
Some	O
authors	O
use	O
the	O
term	O
rather	O
than	O
,	O
but	O
usually	O
the	O
term	O
"	O
Fréchet	B-Algorithm
space	I-Algorithm
"	O
is	O
reserved	O
for	O
locally	B-Algorithm
convex	I-Algorithm
F-spaces	B-Algorithm
.	O
</s>
<s>
Some	O
other	O
authors	O
use	O
the	O
term	O
"	O
F-space	B-Algorithm
"	O
as	O
a	O
synonym	O
of	O
"	O
Fréchet	B-Algorithm
space	I-Algorithm
"	O
,	O
by	O
which	O
they	O
mean	O
a	O
locally	B-Algorithm
convex	I-Algorithm
complete	B-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
metric	O
may	O
or	O
may	O
not	O
necessarily	O
be	O
part	O
of	O
the	O
structure	O
on	O
an	O
F-space	B-Algorithm
;	O
many	O
authors	O
only	O
require	O
that	O
such	O
a	O
space	O
be	O
metrizable	B-Algorithm
in	O
a	O
manner	O
that	O
satisfies	O
the	O
above	O
properties	O
.	O
</s>
<s>
All	O
Banach	O
spaces	O
and	O
Fréchet	B-Algorithm
spaces	I-Algorithm
are	O
F-spaces	B-Algorithm
.	O
</s>
<s>
The	O
Lp	O
spaces	O
can	O
be	O
made	O
into	O
F-spaces	B-Algorithm
for	O
all	O
and	O
for	O
they	O
can	O
be	O
made	O
into	O
locally	B-Algorithm
convex	I-Algorithm
and	O
thus	O
Fréchet	B-Algorithm
spaces	I-Algorithm
and	O
even	O
Banach	O
spaces	O
.	O
</s>
<s>
is	O
an	O
F-space	B-Algorithm
.	O
</s>
<s>
then	O
for	O
are	O
F-spaces	B-Algorithm
under	O
the	O
p-norm	O
:	O
</s>
<s>
The	O
open	O
mapping	O
theorem	O
implies	O
that	O
if	O
are	O
topologies	O
on	O
that	O
make	O
both	O
and	O
into	O
complete	B-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
(	O
for	O
example	O
,	O
Banach	O
or	O
Fréchet	B-Algorithm
spaces	I-Algorithm
)	O
and	O
if	O
one	O
topology	O
is	O
finer	O
or	O
coarser	O
than	O
the	O
other	O
then	O
they	O
must	O
be	O
equal	O
(	O
that	O
is	O
,	O
if	O
)	O
.	O
</s>
<s>
A	O
linear	O
almost	O
continuous	O
map	O
into	O
an	O
F-space	B-Algorithm
whose	O
graph	O
is	O
closed	O
is	O
continuous	O
.	O
</s>
<s>
A	O
linear	O
almost	B-Algorithm
open	I-Algorithm
map	I-Algorithm
into	O
an	O
F-space	B-Algorithm
whose	O
graph	O
is	O
closed	O
is	O
necessarily	O
an	O
open	O
map	O
.	O
</s>
<s>
A	O
linear	O
continuous	O
almost	B-Algorithm
open	I-Algorithm
map	I-Algorithm
from	O
an	O
F-space	B-Algorithm
is	O
necessarily	O
an	O
open	O
map	O
.	O
</s>
<s>
A	O
linear	O
continuous	O
almost	B-Algorithm
open	I-Algorithm
map	I-Algorithm
from	O
an	O
F-space	B-Algorithm
whose	O
image	O
is	O
of	O
the	O
second	O
category	O
in	O
the	O
codomain	O
is	O
necessarily	O
a	O
surjective	B-Algorithm
open	O
map	O
.	O
</s>
