<s>
In	O
the	O
mathematical	O
discipline	O
of	O
functional	B-Application
analysis	I-Application
,	O
a	O
differentiable	O
vector-valued	O
function	O
from	O
Euclidean	O
space	O
is	O
a	O
differentiable	O
function	O
valued	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
whose	O
domains	B-Algorithm
is	O
a	O
subset	O
of	O
some	O
finite-dimensional	O
Euclidean	O
space	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
generalize	O
the	O
notion	O
of	O
derivative	B-Algorithm
to	O
functions	O
whose	O
domain	O
and	O
codomain	O
are	O
subsets	O
of	O
arbitrary	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
in	O
multiple	O
ways	O
.	O
</s>
<s>
But	O
when	O
the	O
domain	O
of	O
a	O
TVS-valued	O
function	O
is	O
a	O
subset	O
of	O
a	O
finite-dimensional	O
Euclidean	O
space	O
then	O
many	O
of	O
these	O
notions	O
become	O
logically	O
equivalent	O
resulting	O
in	O
a	O
much	O
more	O
limited	O
number	O
of	O
generalizations	O
of	O
the	O
derivative	B-Algorithm
and	O
additionally	O
,	O
differentiability	O
is	O
also	O
more	O
well-behaved	O
compared	O
to	O
the	O
general	O
case	O
.	O
</s>
<s>
This	O
article	O
presents	O
the	O
theory	O
of	O
-times	O
continuously	O
differentiable	O
functions	O
on	O
an	O
open	O
subset	O
of	O
Euclidean	O
space	O
(	O
)	O
,	O
which	O
is	O
an	O
important	O
special	O
case	O
of	O
differentiation	B-Algorithm
between	O
arbitrary	O
TVSs	O
.	O
</s>
<s>
This	O
importance	O
stems	O
partially	O
from	O
the	O
fact	O
that	O
every	O
finite-dimensional	O
vector	O
subspace	O
of	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
TVS	O
isomorphic	O
to	O
Euclidean	O
space	O
so	O
that	O
,	O
for	O
example	O
,	O
this	O
special	O
case	O
can	O
be	O
applied	O
to	O
any	O
function	O
whose	O
domain	O
is	O
an	O
arbitrary	O
Hausdorff	O
TVS	O
by	O
restricting	O
it	O
to	O
finite-dimensional	O
vector	O
subspaces	O
.	O
</s>
<s>
TVS-valued	O
)	O
functions	O
which	O
,	O
in	O
particular	O
,	O
are	O
used	O
in	O
the	O
definition	O
of	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
They	O
are	O
fundamental	O
to	O
the	O
analysis	O
of	O
maps	O
between	O
two	O
arbitrary	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
so	O
also	O
to	O
the	O
analysis	O
of	O
TVS-valued	O
maps	O
from	O
Euclidean	O
spaces	O
,	O
which	O
is	O
the	O
focus	O
of	O
this	O
article	O
.	O
</s>
<s>
A	O
continuous	O
map	O
from	O
a	O
subset	O
that	O
is	O
valued	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
said	O
to	O
be	O
(	O
or	O
)	O
if	O
for	O
all	O
it	O
is	O
which	O
by	O
definition	O
means	O
the	O
following	O
limit	O
in	O
exists	O
:	O
</s>
<s>
For	O
any	O
a	O
curve	O
valued	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
called	O
a	O
if	O
it	O
is	O
a	O
topological	O
embedding	O
and	O
a	O
curve	O
such	O
that	O
for	O
every	O
where	O
it	O
is	O
called	O
a	O
if	O
it	O
is	O
also	O
a	O
path	O
(	O
or	O
equivalently	O
,	O
also	O
a	O
-arc	O
)	O
in	O
addition	O
to	O
being	O
a	O
-embedding	O
.	O
</s>
<s>
In	O
this	O
section	O
,	O
the	O
space	O
of	O
smooth	O
test	O
functions	O
and	O
its	O
canonical	O
LF-topology	O
are	O
generalized	O
to	O
functions	O
valued	O
in	O
general	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
(	O
TVSs	O
)	O
.	O
</s>
<s>
Throughout	O
,	O
let	O
be	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
,	O
let	O
and	O
let	O
be	O
either	O
:	O
</s>
<s>
If	O
is	O
metrizable	B-Algorithm
(	O
resp	O
.	O
</s>
<s>
If	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
is	O
a	O
TVS	O
,	O
and	O
is	O
a	O
linear	O
map	O
,	O
then	O
is	O
continuous	O
if	O
and	O
only	O
if	O
for	O
all	O
compact	O
the	O
restriction	O
of	O
to	O
is	O
continuous	O
.	O
</s>
