<s>
In	O
mathematics	O
,	O
the	O
Gateaux	B-Algorithm
differential	I-Algorithm
or	O
Gateaux	B-Algorithm
derivative	I-Algorithm
is	O
a	O
generalization	O
of	O
the	O
concept	O
of	O
directional	O
derivative	O
in	O
differential	O
calculus	O
.	O
</s>
<s>
Named	O
after	O
René	O
Gateaux	O
,	O
a	O
French	O
mathematician	O
who	O
died	O
young	O
in	O
World	O
War	O
I	O
,	O
it	O
is	O
defined	O
for	O
functions	O
between	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
such	O
as	O
Banach	O
spaces	O
.	O
</s>
<s>
Like	O
the	O
Fréchet	O
derivative	O
on	O
a	O
Banach	O
space	O
,	O
the	O
Gateaux	B-Algorithm
differential	I-Algorithm
is	O
often	O
used	O
to	O
formalize	O
the	O
functional	O
derivative	O
commonly	O
used	O
in	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
and	O
physics	O
.	O
</s>
<s>
Unlike	O
other	O
forms	O
of	O
derivatives	O
,	O
the	O
Gateaux	B-Algorithm
differential	I-Algorithm
of	O
a	O
function	O
may	O
be	O
nonlinear	O
.	O
</s>
<s>
However	O
,	O
often	O
the	O
definition	O
of	O
the	O
Gateaux	B-Algorithm
differential	I-Algorithm
also	O
requires	O
that	O
it	O
be	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
Some	O
authors	O
,	O
such	O
as	O
,	O
draw	O
a	O
further	O
distinction	O
between	O
the	O
Gateaux	B-Algorithm
differential	I-Algorithm
(	O
which	O
may	O
be	O
nonlinear	O
)	O
and	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
(	O
which	O
they	O
take	O
to	O
be	O
linear	O
)	O
.	O
</s>
<s>
The	O
limit	O
appearing	O
in	O
(	O
)	O
is	O
taken	O
relative	O
to	O
the	O
topology	O
of	O
If	O
and	O
are	O
real	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
then	O
the	O
limit	O
is	O
taken	O
for	O
real	O
On	O
the	O
other	O
hand	O
,	O
if	O
and	O
are	O
complex	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
then	O
the	O
limit	O
above	O
is	O
usually	O
taken	O
as	O
in	O
the	O
complex	O
plane	O
as	O
in	O
the	O
definition	O
of	O
complex	O
differentiability	O
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
a	O
weak	O
limit	O
is	O
taken	O
instead	O
of	O
a	O
strong	O
limit	O
,	O
which	O
leads	O
to	O
the	O
notion	O
of	O
a	O
weak	O
Gateaux	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
However	O
,	O
this	O
function	O
need	O
not	O
be	O
additive	O
,	O
so	O
that	O
the	O
Gateaux	B-Algorithm
differential	I-Algorithm
may	O
fail	O
to	O
be	O
linear	O
,	O
unlike	O
the	O
Fréchet	O
derivative	O
.	O
</s>
<s>
Furthermore	O
,	O
for	O
Gateaux	B-Algorithm
differentials	I-Algorithm
that	O
linear	O
and	O
continuous	O
in	O
there	O
are	O
several	O
inequivalent	O
ways	O
to	O
formulate	O
their	O
continuous	O
differentiability	O
.	O
</s>
<s>
However	O
this	O
is	O
continuous	O
but	O
not	O
linear	O
in	O
the	O
arguments	O
In	O
infinite	O
dimensions	O
,	O
any	O
discontinuous	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
on	O
is	O
Gateaux	O
differentiable	O
,	O
but	O
its	O
Gateaux	B-Algorithm
differential	I-Algorithm
at	O
is	O
linear	O
but	O
not	O
continuous	O
.	O
</s>
<s>
If	O
is	O
Fréchet	O
differentiable	O
,	O
then	O
it	O
is	O
also	O
Gateaux	O
differentiable	O
,	O
and	O
its	O
Fréchet	O
and	O
Gateaux	B-Algorithm
derivatives	I-Algorithm
agree	O
.	O
</s>
<s>
The	O
converse	O
is	O
clearly	O
not	O
true	O
,	O
since	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
may	O
fail	O
to	O
be	O
linear	O
or	O
continuous	O
.	O
</s>
<s>
In	O
fact	O
,	O
it	O
is	O
even	O
possible	O
for	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
to	O
be	O
linear	O
and	O
continuous	O
but	O
for	O
the	O
Fréchet	O
derivative	O
to	O
fail	O
to	O
exist	O
.	O
</s>
<s>
Nevertheless	O
,	O
for	O
functions	O
from	O
a	O
Banach	O
space	O
to	O
another	O
complex	O
Banach	O
space	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
(	O
where	O
the	O
limit	O
is	O
taken	O
over	O
complex	O
tending	O
to	O
zero	O
as	O
in	O
the	O
definition	O
of	O
complex	O
differentiability	O
)	O
is	O
automatically	O
linear	O
,	O
a	O
theorem	O
of	O
.	O
</s>
<s>
This	O
is	O
analogous	O
to	O
the	O
result	O
from	O
basic	O
complex	O
analysis	O
that	O
a	O
function	O
is	O
analytic	B-Language
if	O
it	O
is	O
complex	O
differentiable	O
in	O
an	O
open	O
set	O
,	O
and	O
is	O
a	O
fundamental	O
result	O
in	O
the	O
study	O
of	O
infinite	O
dimensional	O
holomorphy	O
.	O
</s>
<s>
For	O
instance	O
,	O
differentiation	B-Algorithm
in	I-Algorithm
Fréchet	I-Algorithm
spaces	I-Algorithm
has	O
applications	O
such	O
as	O
the	O
Nash	O
–	O
Moser	O
inverse	O
function	O
theorem	O
in	O
which	O
the	O
function	O
spaces	O
of	O
interest	O
often	O
consist	O
of	O
smooth	O
functions	O
on	O
a	O
manifold	O
.	O
</s>
<s>
Whereas	O
higher	O
order	O
Fréchet	O
derivatives	O
are	O
naturally	O
defined	O
as	O
multilinear	O
functions	O
by	O
iteration	O
,	O
using	O
the	O
isomorphisms	O
higher	O
order	O
Gateaux	B-Algorithm
derivative	I-Algorithm
cannot	O
be	O
defined	O
in	O
this	O
way	O
.	O
</s>
<s>
that	O
arises	O
naturally	O
in	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
as	O
the	O
second	O
variation	O
of	O
at	O
least	O
in	O
the	O
special	O
case	O
where	O
is	O
scalar-valued	O
.	O
</s>
<s>
A	O
version	O
of	O
the	O
fundamental	O
theorem	O
of	O
calculus	O
holds	O
for	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
of	O
provided	O
is	O
assumed	O
to	O
be	O
sufficiently	O
continuously	O
differentiable	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
in	O
the	O
sense	O
that	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
is	O
a	O
continuous	O
function	O
Then	O
for	O
any	O
and	O
where	O
the	O
integral	O
is	O
the	O
Gelfand	B-Algorithm
–	I-Algorithm
Pettis	I-Algorithm
integral	I-Algorithm
(	O
the	O
weak	B-Algorithm
integral	I-Algorithm
)	O
(	O
)	O
.	O
</s>
<s>
(	O
The	O
chain	O
rule	O
)	O
for	O
all	O
and	O
(	O
Importantly	O
,	O
as	O
with	O
simple	O
partial	O
derivatives	O
,	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
does	O
satisfy	O
the	O
chain	O
rule	O
if	O
the	O
derivative	O
is	O
permitted	O
to	O
be	O
discontinuous	O
.	O
)	O
</s>
