<s>
In	O
mathematics	O
,	O
in	O
particular	O
in	O
functional	B-Application
analysis	I-Application
and	O
nonlinear	B-Algorithm
analysis	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
define	O
the	O
derivative	O
of	O
a	O
function	O
between	O
two	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
This	O
notion	O
of	O
differentiation	O
,	O
as	O
it	O
is	O
Gateaux	B-Algorithm
derivative	I-Algorithm
between	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
is	O
significantly	O
weaker	O
than	O
the	O
derivative	O
in	O
a	O
Banach	O
space	O
,	O
even	O
between	O
general	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
With	O
some	O
additional	O
constraints	O
on	O
the	O
Fréchet	B-Algorithm
spaces	I-Algorithm
and	O
functions	O
involved	O
,	O
there	O
is	O
an	O
analog	O
of	O
the	O
inverse	O
function	O
theorem	O
called	O
the	O
Nash	O
–	O
Moser	O
inverse	O
function	O
theorem	O
,	O
having	O
wide	O
applications	O
in	O
nonlinear	B-Algorithm
analysis	I-Algorithm
and	O
differential	B-Language
geometry	I-Language
.	O
</s>
<s>
Formally	O
,	O
the	O
definition	O
of	O
differentiation	O
is	O
identical	O
to	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
Specifically	O
,	O
let	O
and	O
be	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
be	O
an	O
open	O
set	O
,	O
and	O
be	O
a	O
function	O
.	O
</s>
<s>
Let	O
and	O
be	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
The	O
proofs	O
of	O
many	O
of	O
these	O
properties	O
rely	O
fundamentally	O
on	O
the	O
fact	O
that	O
it	O
is	O
possible	O
to	O
define	O
the	O
Riemann	O
integral	O
of	O
continuous	O
curves	O
in	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Surprisingly	O
,	O
a	O
mapping	O
between	O
open	O
subset	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
is	O
smooth	O
(	O
infinitely	O
often	O
differentiable	O
)	O
if	O
it	O
maps	O
smooth	O
curves	O
to	O
smooth	O
curves	O
;	O
see	O
Convenient	O
analysis	O
.	O
</s>
<s>
The	O
existence	O
of	O
a	O
chain	O
rule	O
allows	O
for	O
the	O
definition	O
of	O
a	O
manifold	B-Architecture
modeled	O
on	O
a	O
Frèchet	O
space	O
:	O
a	O
Fréchet	O
manifold	B-Architecture
.	O
</s>
<s>
Furthermore	O
,	O
the	O
linearity	O
of	O
the	O
derivative	O
implies	O
that	O
there	O
is	O
an	O
analog	O
of	O
the	O
tangent	O
bundle	O
for	O
Fréchet	O
manifolds	B-Architecture
.	O
</s>
<s>
Frequently	O
the	O
Fréchet	B-Algorithm
spaces	I-Algorithm
that	O
arise	O
in	O
practical	O
applications	O
of	O
the	O
derivative	O
enjoy	O
an	O
additional	O
property	O
:	O
they	O
are	O
tame	O
.	O
</s>
<s>
Roughly	O
speaking	O
,	O
a	O
tame	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
one	O
which	O
is	O
almost	O
a	O
Banach	O
space	O
.	O
</s>
