<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
named	O
after	O
Maurice	O
Fréchet	O
,	O
are	O
special	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
They	O
are	O
generalizations	O
of	O
Banach	O
spaces	O
(	O
normed	O
vector	O
spaces	O
that	O
are	O
complete	B-Algorithm
with	O
respect	O
to	O
the	O
metric	O
induced	O
by	O
the	O
norm	O
)	O
.	O
</s>
<s>
All	O
Banach	O
and	O
Hilbert	O
spaces	O
are	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Spaces	O
of	O
infinitely	O
differentiable	O
functions	O
are	O
typical	O
examples	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
many	O
of	O
which	O
are	O
typically	O
Banach	O
spaces	O
.	O
</s>
<s>
A	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
defined	O
to	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	B-Architecture
)	O
that	O
is	O
complete	B-Algorithm
as	I-Algorithm
a	I-Algorithm
TVS	I-Algorithm
,	O
meaning	O
that	O
every	O
Cauchy	O
sequence	O
in	O
converges	B-Algorithm
to	O
some	O
point	O
in	O
(	O
see	O
footnote	O
for	O
more	O
details	O
)	O
.	O
</s>
<s>
Important	O
note	O
:	O
Not	O
all	O
authors	O
require	O
that	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
be	O
locally	B-Algorithm
convex	I-Algorithm
(	O
discussed	O
below	O
)	O
.	O
</s>
<s>
The	O
topology	O
of	O
every	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
induced	O
by	O
some	O
translation-invariant	O
complete	B-Algorithm
metric	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
the	O
topology	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
induced	O
by	O
a	O
translation-invariant	O
complete	B-Algorithm
metric	O
then	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Fréchet	O
was	O
the	O
first	O
to	O
use	O
the	O
term	O
"	O
Banach	O
space	O
"	O
and	O
Banach	O
in	O
turn	O
then	O
coined	O
the	O
term	O
"	O
Fréchet	B-Algorithm
space	I-Algorithm
"	O
to	O
mean	O
a	O
complete	B-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
,	O
without	O
the	O
local	O
convexity	O
requirement	O
(	O
such	O
a	O
space	O
is	O
today	O
often	O
called	O
an	O
"	O
F-space	B-Algorithm
"	O
)	O
.	O
</s>
<s>
The	O
condition	O
of	O
locally	B-Algorithm
convex	I-Algorithm
was	O
added	O
later	O
by	O
Nicolas	O
Bourbaki	O
.	O
</s>
<s>
Schaefer	O
)	O
use	O
"	O
F-space	B-Algorithm
"	O
to	O
mean	O
a	O
(	O
locally	B-Algorithm
convex	I-Algorithm
)	O
Fréchet	B-Algorithm
space	I-Algorithm
while	O
others	O
do	O
not	O
require	O
that	O
a	O
"	O
Fréchet	B-Algorithm
space	I-Algorithm
"	O
be	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
some	O
authors	O
even	O
use	O
"	O
F-space	B-Algorithm
"	O
and	O
"	O
Fréchet	B-Algorithm
space	I-Algorithm
"	O
interchangeably	O
.	O
</s>
<s>
When	O
reading	O
mathematical	O
literature	O
,	O
it	O
is	O
recommended	O
that	O
a	O
reader	O
always	O
check	O
whether	O
the	O
book	O
's	O
or	O
article	O
's	O
definition	O
of	O
"	O
-space	O
"	O
and	O
"	O
Fréchet	B-Algorithm
space	I-Algorithm
"	O
requires	O
local	O
convexity	O
.	O
</s>
<s>
Fréchet	B-Algorithm
spaces	I-Algorithm
can	O
be	O
defined	O
in	O
two	O
equivalent	O
ways	O
:	O
the	O
first	O
employs	O
a	O
translation-invariant	O
metric	O
,	O
the	O
second	O
a	O
countable	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
it	O
satisfies	O
the	O
following	O
three	O
properties	O
:	O
</s>
<s>
It	O
is	O
locally	O
convex.Some	O
authors	O
do	O
not	O
include	O
local	O
convexity	O
as	O
part	O
of	O
the	O
definition	O
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Some	O
(	O
or	O
equivalently	O
,	O
every	O
)	O
translation-invariant	O
metric	O
on	O
inducing	O
the	O
topology	O
of	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
Assuming	O
that	O
the	O
other	O
two	O
conditions	O
are	O
satisfied	O
,	O
this	O
condition	O
is	O
equivalent	O
to	O
being	O
a	O
complete	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
,	O
meaning	O
that	O
is	O
a	O
complete	B-Algorithm
uniform	O
space	O
when	O
it	O
is	O
endowed	O
with	O
its	O
canonical	B-Algorithm
uniformity	I-Algorithm
(	O
this	O
canonical	B-Algorithm
uniformity	I-Algorithm
is	O
independent	O
of	O
any	O
metric	O
on	O
and	O
is	O
defined	O
entirely	O
in	O
terms	O
of	O
vector	O
subtraction	O
and	O
'	O
s	O
neighborhoods	O
of	O
the	O
origin	O
;	O
moreover	O
,	O
the	O
uniformity	O
induced	O
by	O
any	O
(	O
topology-defining	O
)	O
translation	O
invariant	O
metric	O
on	O
is	O
identical	O
to	O
this	O
canonical	B-Algorithm
uniformity	I-Algorithm
)	O
.	O
</s>
<s>
Note	O
there	O
is	O
no	O
natural	O
notion	O
of	O
distance	O
between	O
two	O
points	O
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
:	O
many	O
different	O
translation-invariant	O
metrics	O
may	O
induce	O
the	O
same	O
topology	O
.	O
</s>
<s>
The	O
alternative	O
and	O
somewhat	O
more	O
practical	O
definition	O
is	O
the	O
following	O
:	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
it	O
satisfies	O
the	O
following	O
three	O
properties	O
:	O
</s>
<s>
it	O
is	O
complete	B-Algorithm
with	O
respect	O
to	O
the	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
A	O
sequence	O
in	O
converges	B-Algorithm
to	O
in	O
the	O
Fréchet	B-Algorithm
space	I-Algorithm
defined	O
by	O
a	O
family	O
of	O
seminorms	O
if	O
and	O
only	O
if	O
it	O
converges	B-Algorithm
to	O
with	O
respect	O
to	O
each	O
of	O
the	O
given	O
seminorms	O
.	O
</s>
<s>
In	O
contrast	O
to	O
Banach	O
spaces	O
,	O
the	O
complete	B-Algorithm
translation-invariant	O
metric	O
need	O
not	O
arise	O
from	O
a	O
norm	O
.	O
</s>
<s>
The	O
topology	O
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
does	O
,	O
however	O
,	O
arise	O
from	O
both	O
a	O
total	O
paranorm	O
and	O
an	O
-norm	O
(	O
the	O
stands	O
for	O
Fréchet	O
)	O
.	O
</s>
<s>
Even	O
though	O
the	O
topological	O
structure	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
is	O
more	O
complicated	O
than	O
that	O
of	O
Banach	O
spaces	O
due	O
to	O
the	O
potential	O
lack	O
of	O
a	O
norm	O
,	O
many	O
important	O
results	O
in	O
functional	B-Application
analysis	I-Application
,	O
like	O
the	O
open	O
mapping	O
theorem	O
,	O
the	O
closed	O
graph	O
theorem	O
,	O
and	O
the	O
Banach	O
–	O
Steinhaus	O
theorem	O
,	O
still	O
hold	O
.	O
</s>
<s>
However	O
,	O
seminorms	O
are	O
useful	O
in	O
that	O
they	O
enable	O
us	O
to	O
construct	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
as	O
follows	O
:	O
</s>
<s>
To	O
construct	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
one	O
typically	O
starts	O
with	O
a	O
vector	O
space	O
and	O
defines	O
a	O
countable	O
family	O
of	O
seminorms	O
on	O
with	O
the	O
following	O
two	O
properties	O
:	O
</s>
<s>
Then	O
the	O
topology	O
induced	O
by	O
these	O
seminorms	O
(	O
as	O
explained	O
above	O
)	O
turns	O
into	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
;	O
the	O
first	O
property	O
ensures	O
that	O
it	O
is	O
Hausdorff	O
,	O
and	O
the	O
second	O
property	O
ensures	O
that	O
it	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
Every	O
Banach	O
space	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
as	O
the	O
norm	O
induces	O
a	O
translation-invariant	O
metric	O
and	O
the	O
space	O
is	O
complete	B-Algorithm
with	O
respect	O
to	O
this	O
metric	O
.	O
</s>
<s>
The	O
space	O
of	O
all	O
real	B-Algorithm
valued	I-Algorithm
sequences	I-Algorithm
becomes	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
we	O
define	O
the	O
-th	O
seminorm	O
of	O
a	O
sequence	O
to	O
be	O
the	O
absolute	O
value	O
of	O
the	O
-th	O
element	O
of	O
the	O
sequence	O
.	O
</s>
<s>
Convergence	B-Algorithm
in	O
this	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
equivalent	O
to	O
element-wise	O
convergence	B-Algorithm
.	O
</s>
<s>
for	O
every	O
non-negative	O
integer	O
Here	O
,	O
denotes	O
the	O
-th	O
derivative	O
of	O
and	O
In	O
this	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
a	O
sequence	O
of	O
functions	O
converges	B-Algorithm
towards	O
the	O
element	O
if	O
and	O
only	O
if	O
for	O
every	O
non-negative	O
integer	O
the	O
sequence	O
converges	B-Algorithm
uniformly	I-Algorithm
.	O
</s>
<s>
for	O
all	O
integers	O
Then	O
,	O
a	O
sequence	O
of	O
functions	O
converges	B-Algorithm
if	O
and	O
only	O
if	O
for	O
every	O
the	O
sequences	O
converge	B-Algorithm
compactly	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
compact	O
-manifold	O
and	O
is	O
a	O
Banach	O
space	O
,	O
then	O
the	O
set	O
of	O
all	O
infinitely-often	O
differentiable	O
functions	O
can	O
be	O
turned	O
into	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
by	O
using	O
as	O
seminorms	O
the	O
suprema	O
of	O
the	O
norms	O
of	O
all	O
partial	O
derivatives	O
.	O
</s>
<s>
If	O
is	O
a	O
(	O
not	O
necessarily	O
compact	O
)	O
-manifold	O
which	O
admits	O
a	O
countable	O
sequence	O
of	O
compact	O
subsets	O
,	O
so	O
that	O
every	O
compact	O
subset	O
of	O
is	O
contained	O
in	O
at	O
least	O
one	O
then	O
the	O
spaces	O
and	O
are	O
also	O
Fréchet	B-Algorithm
space	I-Algorithm
in	O
a	O
natural	O
manner	O
.	O
</s>
<s>
(	O
where	O
is	O
the	O
norm	O
induced	O
by	O
the	O
Riemannian	O
metric	O
)	O
is	O
a	O
family	O
of	O
seminorms	O
making	O
into	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
makes	O
into	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
makes	O
into	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Not	O
all	O
vector	O
spaces	O
with	O
complete	B-Algorithm
translation-invariant	O
metrics	O
are	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Although	O
this	O
space	O
fails	O
to	O
be	O
locally	B-Algorithm
convex	I-Algorithm
,	O
it	O
is	O
an	O
F-space	B-Algorithm
.	O
</s>
<s>
If	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
admits	O
a	O
continuous	O
norm	O
then	O
all	O
of	O
the	O
seminorms	O
used	O
to	O
define	O
it	O
can	O
be	O
replaced	O
with	O
norms	O
by	O
adding	O
this	O
continuous	O
norm	O
to	O
each	O
of	O
them	O
.	O
</s>
<s>
A	O
closed	O
subspace	O
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
quotient	O
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
by	O
a	O
closed	O
subspace	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
direct	O
sum	O
of	O
a	O
finite	O
number	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
product	O
of	O
countably	O
many	O
Fréchet	B-Algorithm
spaces	I-Algorithm
is	O
always	O
once	O
again	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
However	O
,	O
an	O
arbitrary	O
product	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
will	O
be	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
all	O
for	O
at	O
most	O
countably	O
many	O
of	O
them	O
are	O
trivial	O
(	O
that	O
is	O
,	O
have	O
dimension	O
0	O
)	O
.	O
</s>
<s>
Consequently	O
,	O
a	O
product	O
of	O
uncountably	O
many	O
non-trivial	O
Fréchet	B-Algorithm
spaces	I-Algorithm
can	O
not	O
be	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
(	O
indeed	O
,	O
such	O
a	O
product	O
is	O
not	O
even	O
metrizable	B-Algorithm
because	O
its	O
origin	O
can	O
not	O
have	O
a	O
countable	O
neighborhood	O
basis	O
)	O
.	O
</s>
<s>
So	O
for	O
example	O
,	O
if	O
is	O
any	O
set	O
and	O
is	O
any	O
non-trivial	O
Fréchet	B-Algorithm
space	I-Algorithm
(	O
such	O
as	O
for	O
instance	O
)	O
,	O
then	O
the	O
product	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
is	O
a	O
countable	O
set	O
.	O
</s>
<s>
Several	O
important	O
tools	O
of	O
functional	B-Application
analysis	I-Application
which	O
are	O
based	O
on	O
the	O
Baire	O
category	O
theorem	O
remain	O
true	O
in	O
Fréchet	B-Algorithm
spaces	I-Algorithm
;	O
examples	O
are	O
the	O
closed	O
graph	O
theorem	O
and	O
the	O
open	O
mapping	O
theorem	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
TVSs	I-Algorithm
(	O
such	O
as	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>
Every	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
from	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
into	O
another	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	B-Architecture
)	O
is	O
continuous	O
.	O
</s>
<s>
All	O
Fréchet	B-Algorithm
spaces	I-Algorithm
are	O
stereotype	O
spaces	O
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
stereotype	O
spaces	O
Fréchet	B-Algorithm
spaces	I-Algorithm
are	O
dual	O
objects	O
to	O
Brauner	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
All	O
metrizable	B-Algorithm
Montel	B-Algorithm
spaces	I-Algorithm
are	O
separable	O
.	O
</s>
<s>
A	O
separable	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
each	O
weak-*	O
convergent	O
sequence	O
in	O
its	O
continuous	O
dual	O
converges	B-Algorithm
is	O
strongly	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
(	O
and	O
more	O
generally	O
,	O
of	O
any	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-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
and	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
a	O
Ptak	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	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Even	O
if	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
has	O
a	O
topology	O
that	O
is	O
defined	O
by	O
a	O
(	O
countable	O
)	O
family	O
of	O
(	O
all	O
norms	O
are	O
also	O
seminorms	O
)	O
,	O
then	O
it	O
may	O
nevertheless	O
still	O
fail	O
to	O
be	O
normable	O
space	O
(	O
meaning	O
that	O
its	O
topology	O
can	O
not	O
be	O
defined	O
by	O
any	O
single	O
norm	O
)	O
.	O
</s>
<s>
The	O
space	O
of	O
all	O
sequences	O
(	O
with	O
the	O
product	O
topology	O
)	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
There	O
does	O
not	O
exist	O
any	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
that	O
is	O
strictly	O
coarser	O
than	O
this	O
product	O
topology	O
.	O
</s>
<s>
Also	O
,	O
there	O
does	O
not	O
exist	O
continuous	O
norm	O
on	O
In	O
fact	O
,	O
as	O
the	O
following	O
theorem	O
shows	O
,	O
whenever	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
on	O
which	O
there	O
does	O
not	O
exist	O
any	O
continuous	O
norm	O
,	O
then	O
this	O
is	O
due	O
entirely	O
to	O
the	O
presence	O
of	O
as	O
a	O
subspace	O
.	O
</s>
<s>
If	O
is	O
a	O
non-normable	O
Fréchet	B-Algorithm
space	I-Algorithm
on	O
which	O
there	O
exists	O
a	O
continuous	O
norm	O
,	O
then	O
contains	O
a	O
closed	O
vector	O
subspace	O
that	O
has	O
no	O
topological	O
complement	O
.	O
</s>
<s>
A	O
metrizable	B-Algorithm
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>
In	O
particular	O
,	O
if	O
a	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
space	O
(	O
such	O
as	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
)	O
is	O
normable	O
(	O
which	O
can	O
only	O
happen	O
if	O
is	O
infinite	O
dimensional	O
)	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	B-Algorithm
nor	O
normable	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
(	O
and	O
more	O
generally	O
,	O
of	O
bornological	B-Algorithm
spaces	I-Algorithm
such	O
as	O
metrizable	B-Algorithm
TVSs	I-Algorithm
)	O
is	O
always	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
and	O
so	O
like	O
any	O
complete	B-Algorithm
TVS	I-Algorithm
,	O
it	O
is	O
normable	O
if	O
and	O
only	O
if	O
its	O
topology	O
can	O
be	O
induced	O
by	O
a	O
complete	B-Algorithm
norm	O
(	O
that	O
is	O
,	O
if	O
and	O
only	O
if	O
it	O
can	O
be	O
made	O
into	O
a	O
Banach	O
space	O
that	O
has	O
the	O
same	O
topology	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
then	O
is	O
normable	O
if	O
(	O
and	O
only	O
if	O
)	O
there	O
exists	O
a	O
complete	B-Algorithm
norm	O
on	O
its	O
continuous	O
dual	O
space	O
such	O
that	O
the	O
norm	O
induced	O
topology	O
on	O
is	O
finer	O
than	O
the	O
weak-*	O
topology	O
.	O
</s>
<s>
Consequently	O
,	O
if	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
normable	O
(	O
which	O
can	O
only	O
happen	O
if	O
it	O
is	O
infinite	O
dimensional	O
)	O
then	O
neither	O
is	O
its	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
then	O
the	O
space	O
consisting	O
of	O
all	O
continuous	O
linear	B-Architecture
maps	I-Architecture
from	O
to	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
in	O
any	O
natural	O
manner	O
.	O
</s>
<s>
This	O
is	O
a	O
major	O
difference	O
between	O
the	O
theory	O
of	O
Banach	O
spaces	O
and	O
that	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
and	O
necessitates	O
a	O
different	O
definition	O
for	O
continuous	O
differentiability	O
of	O
functions	O
defined	O
on	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
:	O
</s>
<s>
Since	O
the	O
product	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
is	O
again	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
we	O
can	O
then	O
try	O
to	O
differentiate	O
and	O
define	O
the	O
higher	O
derivatives	O
of	O
in	O
this	O
fashion	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
inverse	O
function	O
theorem	O
is	O
not	O
true	O
in	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
although	O
a	O
partial	O
substitute	O
is	O
the	O
Nash	O
–	O
Moser	O
theorem	O
.	O
</s>
<s>
One	O
may	O
define	O
Fréchet	O
manifolds	B-Architecture
as	O
spaces	O
that	O
"	O
locally	O
look	O
like	O
"	O
Fréchet	B-Algorithm
spaces	I-Algorithm
(	O
just	O
like	O
ordinary	O
manifolds	B-Architecture
are	O
defined	O
as	O
spaces	O
that	O
locally	O
look	O
like	O
Euclidean	O
space	O
)	O
,	O
and	O
one	O
can	O
then	O
extend	O
the	O
concept	O
of	O
Lie	O
group	O
to	O
these	O
manifolds	B-Architecture
.	O
</s>
<s>
If	O
we	O
drop	O
the	O
requirement	O
for	O
the	O
space	O
to	O
be	O
locally	B-Algorithm
convex	I-Algorithm
,	O
we	O
obtain	O
F-spaces	B-Algorithm
:	O
vector	O
spaces	O
with	O
complete	B-Algorithm
translation-invariant	O
metrics	O
.	O
</s>
<s>
LF-spaces	B-Algorithm
are	O
countable	O
inductive	O
limits	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
