<s>
In	O
mathematics	O
,	O
Schwartz	B-Algorithm
space	I-Algorithm
is	O
the	O
function	B-Algorithm
space	I-Algorithm
of	O
all	O
functions	O
whose	O
derivatives	B-Algorithm
are	O
rapidly	O
decreasing	O
.	O
</s>
<s>
This	O
space	O
has	O
the	O
important	O
property	O
that	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
an	O
automorphism	O
on	O
this	O
space	O
.	O
</s>
<s>
This	O
property	O
enables	O
one	O
,	O
by	O
duality	O
,	O
to	O
define	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
for	O
elements	O
in	O
the	O
dual	O
space	O
of	O
,	O
that	O
is	O
,	O
for	O
tempered	O
distributions	O
.	O
</s>
<s>
A	O
function	O
in	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
is	O
sometimes	O
called	O
a	O
Schwartz	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Schwartz	B-Algorithm
space	I-Algorithm
is	O
named	O
after	O
French	O
mathematician	O
Laurent	O
Schwartz	O
.	O
</s>
<s>
The	O
Schwartz	B-Algorithm
space	I-Algorithm
or	O
space	O
of	O
rapidly	O
decreasing	O
functions	O
on	O
is	O
the	O
function	O
spacewhere	O
is	O
the	O
function	B-Algorithm
space	I-Algorithm
of	O
smooth	O
functions	O
from	O
into	O
,	O
and	O
Here	O
,	O
denotes	O
the	O
supremum	O
,	O
and	O
we	O
used	O
multi-index	O
notation	O
,	O
i.e.	O
</s>
<s>
In	O
particular	O
,	O
(	O
,	O
)	O
is	O
a	O
subspace	O
of	O
the	O
function	B-Algorithm
space	I-Algorithm
(	O
,	O
)	O
of	O
smooth	O
functions	O
from	O
into	O
.	O
</s>
<s>
This	O
is	O
clear	O
since	O
any	O
derivative	B-Algorithm
of	O
f	O
is	O
continuous	O
and	O
supported	O
in	O
the	O
support	O
of	O
f	O
,	O
so	O
(	O
xαDβ	O
)	O
f	O
has	O
a	O
maximum	O
in	O
Rn	O
by	O
the	O
extreme	O
value	O
theorem	O
.	O
</s>
<s>
Because	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
is	O
a	O
vector	O
space	O
,	O
any	O
polynomial	O
can	O
by	O
multiplied	O
by	O
a	O
factor	O
for	O
a	O
real	O
constant	O
,	O
to	O
give	O
an	O
element	O
of	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
there	O
is	O
an	O
embedding	O
of	O
polynomials	O
inside	O
a	O
Schwartz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
a	O
linear	B-Architecture
isomorphism	I-Architecture
.	O
</s>
<s>
is	O
a	O
distinguished	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
Fréchet	B-Algorithm
Schwartz	B-Algorithm
TVS	I-Algorithm
over	O
the	O
complex	O
numbers	O
.	O
</s>
<s>
Both	O
and	O
its	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
are	O
also	O
:	O
</s>
<s>
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
,	O
</s>
<s>
nuclear	B-Algorithm
Montel	B-Algorithm
spaces	I-Algorithm
,	O
</s>
<s>
It	O
is	O
known	O
that	O
in	O
the	O
dual	O
space	O
of	O
any	O
Montel	B-Algorithm
space	I-Algorithm
,	O
a	O
sequence	O
converges	O
in	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
if	O
and	O
only	O
if	O
it	O
converges	O
in	O
the	O
weak*	O
topology	O
,	O
</s>
<s>
Ultrabornological	B-Algorithm
spaces	I-Algorithm
,	O
</s>
<s>
reflexive	O
barrelled	B-Algorithm
Mackey	B-Algorithm
spaces	I-Algorithm
.	O
</s>
