<s>
In	O
mathematical	O
analysis	O
,	O
the	O
spaces	B-Algorithm
of	I-Algorithm
test	I-Algorithm
functions	I-Algorithm
and	I-Algorithm
distributions	I-Algorithm
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
that	O
are	O
used	O
in	O
the	O
definition	O
and	O
application	O
of	O
distributions	O
.	O
</s>
<s>
The	O
space	O
of	O
all	O
test	O
functions	O
,	O
denoted	O
by	O
is	O
endowed	O
with	O
a	O
certain	O
topology	O
,	O
called	O
the	O
,	O
that	O
makes	O
into	O
a	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	B-Architecture
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
is	O
called	O
and	O
is	O
denoted	O
by	O
where	O
the	O
""	O
subscript	O
indicates	O
that	O
the	O
continuous	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
denoted	O
by	O
is	O
endowed	O
with	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
If	O
then	O
the	O
use	O
of	O
Schwartz	B-Algorithm
functions	I-Algorithm
as	O
test	O
functions	O
gives	O
rise	O
to	O
a	O
certain	O
subspace	O
of	O
whose	O
elements	O
are	O
called	O
.	O
</s>
<s>
These	O
are	O
important	O
because	O
they	O
allow	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
be	O
extended	O
from	O
"	O
standard	O
functions	O
"	O
to	O
tempered	O
distributions	O
.	O
</s>
<s>
There	O
also	O
exist	O
other	O
major	O
classes	O
of	O
test	O
functions	O
that	O
are	O
subsets	O
of	O
such	O
as	O
spaces	O
of	O
analytic	B-Language
test	I-Language
functions	I-Language
,	O
which	O
produce	O
very	O
different	O
classes	O
of	O
distributions	O
.	O
</s>
<s>
The	O
theory	O
of	O
such	O
distributions	O
has	O
a	O
different	O
character	O
from	O
the	O
previous	O
one	O
because	O
there	O
are	O
no	O
analytic	B-Language
functions	I-Language
with	O
non-empty	O
compact	O
support	O
.	O
</s>
<s>
Use	O
of	O
analytic	B-Language
test	I-Language
functions	I-Language
leads	O
to	O
Sato	O
's	O
theory	O
of	O
hyperfunctions	O
.	O
</s>
<s>
For	O
two	O
functions	O
,	O
the	O
following	O
notation	O
defines	O
a	O
canonical	O
pairing	B-Algorithm
:	O
</s>
<s>
Distributions	O
on	O
are	O
defined	O
to	O
be	O
the	O
continuous	B-Algorithm
linear	B-Algorithm
functionals	I-Algorithm
on	O
when	O
this	O
vector	O
space	O
is	O
endowed	O
with	O
a	O
particular	O
topology	O
called	O
the	O
.	O
</s>
<s>
Proposition	O
:	O
If	O
is	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
then	O
the	O
is	O
a	O
distribution	O
if	O
and	O
only	O
if	O
the	O
following	O
equivalent	O
conditions	O
are	O
satisfied	O
:	O
</s>
<s>
To	O
define	O
the	O
space	O
of	O
distributions	O
we	O
must	O
first	O
define	O
the	O
canonical	O
LF-topology	O
,	O
which	O
in	O
turn	O
requires	O
that	O
several	O
other	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
(	O
TVSs	O
)	O
be	O
defined	O
first	O
.	O
</s>
<s>
First	O
,	O
a	O
(	O
non-normable	O
)	O
topology	O
on	O
will	O
be	O
defined	O
,	O
then	O
every	O
will	O
be	O
endowed	O
with	O
the	O
subspace	O
topology	O
induced	O
on	O
it	O
by	O
and	O
finally	O
the	O
(	O
non-metrizable	O
)	O
canonical	O
LF-topology	O
on	O
will	O
be	O
defined	O
.	O
</s>
<s>
The	O
space	O
of	O
distributions	O
,	O
being	O
defined	O
as	O
the	O
continuous	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
is	O
then	O
endowed	O
with	O
the	O
(	O
non-metrizable	O
)	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
induced	O
by	O
and	O
the	O
canonical	O
LF-topology	O
(	O
this	O
topology	O
is	O
a	O
generalization	O
of	O
the	O
usual	O
operator	O
norm	O
induced	O
topology	O
that	O
is	O
placed	O
on	O
the	O
continuous	B-Algorithm
dual	O
spaces	O
of	O
normed	O
spaces	O
)	O
.	O
</s>
<s>
All	O
of	O
the	O
functions	O
above	O
are	O
non-negative	O
-valued	O
seminorms	O
on	O
As	O
explained	O
in	O
this	O
article	O
,	O
every	O
set	O
of	O
seminorms	O
on	O
a	O
vector	O
space	O
induces	O
a	O
locally	B-Algorithm
convex	I-Algorithm
vector	B-Architecture
topology	I-Architecture
.	O
</s>
<s>
generate	O
the	O
same	O
locally	B-Algorithm
convex	I-Algorithm
vector	B-Architecture
topology	I-Architecture
on	O
(	O
so	O
for	O
example	O
,	O
the	O
topology	O
generated	O
by	O
the	O
seminorms	O
in	O
is	O
equal	O
to	O
the	O
topology	O
generated	O
by	O
those	O
in	O
)	O
.	O
</s>
<s>
With	O
this	O
topology	O
,	O
becomes	O
a	O
locally	B-Algorithm
convex	I-Algorithm
Fréchet	B-Algorithm
space	I-Algorithm
that	O
is	O
normable	O
.	O
</s>
<s>
If	O
the	O
family	O
of	O
compact	O
sets	O
satisfies	O
and	O
for	O
all	O
then	O
a	O
complete	B-Algorithm
translation-invariant	O
metric	O
on	O
can	O
be	O
obtained	O
by	O
taking	O
a	O
suitable	O
countable	O
Fréchet	O
combination	O
of	O
any	O
one	O
of	O
the	O
above	O
families	O
.	O
</s>
<s>
For	O
any	O
compact	O
subset	O
is	O
a	O
closed	O
subspace	O
of	O
the	O
Fréchet	B-Algorithm
space	I-Algorithm
and	O
is	O
thus	O
also	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
For	O
all	O
compact	O
satisfying	O
denote	O
the	O
inclusion	B-Algorithm
map	I-Algorithm
by	O
Then	O
this	O
map	O
is	O
a	O
linear	O
embedding	O
of	O
TVSs	O
(	O
that	O
is	O
,	O
it	O
is	O
a	O
linear	O
map	O
that	O
is	O
also	O
a	O
topological	O
embedding	O
)	O
whose	O
image	O
(	O
or	O
"	O
range	O
"	O
)	O
is	O
closed	O
in	O
its	O
codomain	B-Algorithm
;	O
said	O
differently	O
,	O
the	O
topology	O
on	O
is	O
identical	O
to	O
the	O
subspace	O
topology	O
it	O
inherits	O
from	O
and	O
also	O
is	O
a	O
closed	O
subset	O
of	O
The	O
interior	O
of	O
relative	O
to	O
is	O
empty	O
.	O
</s>
<s>
The	O
space	O
is	O
a	O
distinguished	B-Algorithm
Schwartz	B-Algorithm
Montel	B-Algorithm
space	I-Algorithm
so	O
if	O
then	O
it	O
is	O
normable	O
and	O
thus	O
a	O
Banach	O
space	O
(	O
although	O
like	O
all	O
other	O
it	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
If	O
is	O
restricted	O
to	O
then	O
the	O
following	O
induced	O
linear	O
map	O
is	O
a	O
homeomorphism	O
(	O
and	O
thus	O
a	O
TVS-isomorphism	O
)	O
:	O
</s>
<s>
the	O
vector	O
space	O
is	O
canonically	O
identified	O
with	O
its	O
image	O
in	O
(	O
however	O
,	O
if	O
then	O
is	O
a	O
topological	O
embedding	O
when	O
these	O
spaces	O
are	O
endowed	O
with	O
their	O
canonical	O
LF	O
topologies	O
,	O
although	O
it	O
is	O
continuous	B-Algorithm
)	O
.	O
</s>
<s>
forms	O
a	O
direct	O
system	O
in	O
the	O
category	O
of	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
that	O
is	O
directed	O
by	O
(	O
under	O
subset	O
inclusion	O
)	O
.	O
</s>
<s>
This	O
system	O
's	O
direct	O
limit	O
(	O
in	O
the	O
category	O
of	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
)	O
is	O
the	O
pair	O
where	O
are	O
the	O
natural	O
inclusions	O
and	O
where	O
is	O
now	O
endowed	O
with	O
the	O
(	O
unique	O
)	O
strongest	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
making	O
all	O
of	O
the	O
inclusion	B-Algorithm
maps	I-Algorithm
continuous	B-Algorithm
.	O
</s>
<s>
Note	O
that	O
any	O
convex	O
set	O
satisfying	O
this	O
condition	O
is	O
necessarily	O
absorbing	B-Algorithm
in	O
Since	O
the	O
topology	O
of	O
any	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
translation-invariant	O
,	O
any	O
TVS-topology	O
is	O
completely	O
determined	O
by	O
the	O
set	O
of	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
In	O
particular	O
,	O
we	O
may	O
deduce	O
that	O
the	O
canonical	O
LF	O
topology	O
makes	O
into	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
strict	B-Algorithm
LF-space	I-Algorithm
(	O
and	O
also	O
a	O
strict	B-Algorithm
LB-space	I-Algorithm
if	O
)	O
,	O
which	O
of	O
course	O
is	O
the	O
reason	O
why	O
this	O
topology	O
is	O
called	O
"	O
the	O
canonical	O
LF	O
topology	O
"	O
(	O
see	O
this	O
footnote	O
for	O
more	O
details	O
)	O
.	O
</s>
<s>
From	O
the	O
universal	O
property	O
of	O
direct	O
limits	O
,	O
we	O
know	O
that	O
if	O
is	O
a	O
linear	O
map	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
(	O
not	O
necessarily	O
Hausdorff	O
)	O
,	O
then	O
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
is	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
for	O
every	O
the	O
restriction	O
of	O
to	O
is	O
continuous	B-Algorithm
(	O
or	O
bounded	B-Algorithm
)	O
.	O
</s>
<s>
This	O
map	O
is	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Consequently	O
,	O
if	O
then	O
the	O
transpose	B-Algorithm
of	O
is	O
neither	O
one-to-one	O
nor	O
onto	O
.	O
</s>
<s>
It	O
follows	O
from	O
Baire	O
's	O
theorem	O
that	O
is	O
metrizable	B-Algorithm
and	O
thus	O
also	O
normable	O
(	O
see	O
this	O
footnote	O
for	O
an	O
explanation	O
of	O
how	O
the	O
non-metrizable	O
space	O
can	O
be	O
complete	B-Algorithm
even	O
though	O
it	O
does	O
not	O
admit	O
a	O
metric	O
)	O
.	O
</s>
<s>
The	O
fact	O
that	O
is	O
a	O
nuclear	B-Algorithm
Montel	B-Algorithm
space	I-Algorithm
makes	O
up	O
for	O
the	O
non-metrizability	O
of	O
(	O
see	O
this	O
footnote	O
for	O
a	O
more	O
detailed	O
explanation	O
)	O
.	O
</s>
<s>
Said	O
differently	O
,	O
the	O
topology	O
on	O
is	O
identical	O
to	O
the	O
subspace	O
topology	O
that	O
it	O
inherits	O
from	O
where	O
recall	O
that	O
'	O
s	O
topology	O
was	O
to	O
be	O
the	O
subspace	O
topology	O
induced	O
on	O
it	O
by	O
In	O
particular	O
,	O
both	O
and	O
induces	O
the	O
same	O
subspace	O
topology	O
on	O
However	O
,	O
this	O
does	O
imply	O
that	O
the	O
canonical	O
LF	O
topology	O
on	O
is	O
equal	O
to	O
the	O
subspace	O
topology	O
induced	O
on	O
by	O
;	O
these	O
two	O
topologies	O
on	O
are	O
in	O
fact	O
equal	O
to	O
each	O
other	O
since	O
the	O
canonical	O
LF	O
topology	O
is	O
metrizable	B-Algorithm
while	O
the	O
subspace	O
topology	O
induced	O
on	O
it	O
by	O
is	O
metrizable	B-Algorithm
(	O
since	O
recall	O
that	O
is	O
metrizable	B-Algorithm
)	O
.	O
</s>
<s>
The	O
canonical	O
LF	O
topology	O
on	O
is	O
actually	O
than	O
the	O
subspace	O
topology	O
that	O
it	O
inherits	O
from	O
(	O
thus	O
the	O
natural	O
inclusion	O
is	O
continuous	B-Algorithm
but	O
a	O
topological	O
embedding	O
)	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
canonical	O
LF	O
topology	O
is	O
so	O
fine	O
that	O
if	O
denotes	O
some	O
linear	O
map	O
that	O
is	O
a	O
"	O
natural	O
inclusion	O
"	O
(	O
such	O
as	O
or	O
or	O
other	O
maps	O
discussed	O
below	O
)	O
then	O
this	O
map	O
will	O
typically	O
be	O
continuous	B-Algorithm
,	O
which	O
as	O
is	O
shown	O
below	O
,	O
is	O
ultimately	O
the	O
reason	O
why	O
locally	O
integrable	O
functions	O
,	O
Radon	O
measures	O
,	O
etc	O
.	O
</s>
<s>
all	O
induce	O
distributions	O
(	O
via	O
the	O
transpose	B-Algorithm
of	O
such	O
a	O
"	O
natural	O
inclusion	O
"	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
since	O
distributions	O
are	O
just	O
continuous	B-Algorithm
linear	B-Algorithm
functionals	I-Algorithm
on	O
the	O
fine	O
nature	O
of	O
the	O
canonical	O
LF	O
topology	O
means	O
that	O
more	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
end	O
up	O
being	O
continuous	B-Algorithm
(	O
"	O
more	O
"	O
means	O
as	O
compared	O
to	O
a	O
coarser	O
topology	O
that	O
we	O
could	O
have	O
placed	O
on	O
such	O
as	O
for	O
instance	O
,	O
the	O
subspace	O
topology	O
induced	O
by	O
some	O
which	O
although	O
it	O
would	O
have	O
made	O
metrizable	B-Algorithm
,	O
it	O
would	O
have	O
also	O
resulted	O
in	O
fewer	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
being	O
continuous	B-Algorithm
and	O
thus	O
there	O
would	O
have	O
been	O
fewer	O
distributions	O
;	O
moreover	O
,	O
this	O
particular	O
coarser	O
topology	O
also	O
has	O
the	O
disadvantage	O
of	O
not	O
making	O
into	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
)	O
.	O
</s>
<s>
The	O
differentiation	O
map	O
is	O
a	O
surjective	B-Algorithm
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
The	O
bilinear	O
multiplication	O
map	O
given	O
by	O
is	O
continuous	B-Algorithm
;	O
it	O
is	O
however	O
,	O
hypocontinuous	B-Algorithm
.	O
</s>
<s>
As	O
discussed	O
earlier	O
,	O
continuous	B-Algorithm
linear	B-Algorithm
functionals	I-Algorithm
on	O
a	O
are	O
known	O
as	O
distributions	O
on	O
.	O
</s>
<s>
If	O
is	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
:	O
is	O
a	O
continuous	B-Algorithm
function	O
.	O
</s>
<s>
is	O
continuous	B-Algorithm
at	O
the	O
origin	O
.	O
</s>
<s>
is	O
uniformly	B-Algorithm
continuous	B-Algorithm
.	O
</s>
<s>
is	O
a	O
bounded	B-Algorithm
operator	O
.	O
</s>
<s>
is	O
sequentially	O
continuous	B-Algorithm
.	O
</s>
<s>
is	O
sequentially	O
continuous	B-Algorithm
at	O
the	O
origin	O
;	O
in	O
other	O
words	O
,	O
maps	O
null	O
sequences	O
to	O
null	O
sequences	O
.	O
</s>
<s>
maps	O
null	O
sequences	O
to	O
bounded	B-Algorithm
subsets	I-Algorithm
.	O
</s>
<s>
explicitly	O
,	O
for	O
every	O
sequence	O
in	O
that	O
converges	O
in	O
to	O
the	O
origin	O
,	O
the	O
sequence	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
maps	O
Mackey	O
convergent	O
null	O
sequences	O
to	O
bounded	B-Algorithm
subsets	I-Algorithm
;	O
</s>
<s>
explicitly	O
,	O
for	O
every	O
Mackey	O
convergent	O
null	O
sequence	O
in	O
the	O
sequence	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
a	O
sequence	O
is	O
said	O
to	O
be	O
if	O
there	O
exists	O
a	O
divergent	O
sequence	O
of	O
positive	O
real	O
number	O
such	O
that	O
the	O
sequence	O
is	O
bounded	B-Algorithm
;	O
every	O
sequence	O
that	O
is	O
Mackey	O
convergent	O
to	O
necessarily	O
converges	O
to	O
the	O
origin	O
(	O
in	O
the	O
usual	O
sense	O
)	O
.	O
</s>
<s>
The	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	I-Algorithm
bounded	I-Algorithm
subsets	I-Algorithm
is	O
also	O
called	O
.	O
</s>
<s>
This	O
topology	O
is	O
chosen	O
because	O
it	O
is	O
with	O
this	O
topology	O
that	O
becomes	O
a	O
nuclear	B-Algorithm
Montel	B-Algorithm
space	I-Algorithm
and	O
it	O
is	O
with	O
this	O
topology	O
that	O
the	O
kernels	O
theorem	O
of	O
Schwartz	B-Algorithm
holds	O
.	O
</s>
<s>
No	O
matter	O
which	O
topology	O
is	O
chosen	O
,	O
will	O
be	O
a	O
non-metrizable	O
,	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
space	O
is	O
separable	O
and	O
has	O
the	O
strong	O
Pytkeev	O
property	O
but	O
it	O
is	O
neither	O
a	O
k-space	O
nor	O
a	O
sequential	O
space	O
,	O
which	O
in	O
particular	O
implies	O
that	O
it	O
is	O
not	O
metrizable	B-Algorithm
and	O
also	O
that	O
its	O
topology	O
can	O
be	O
defined	O
using	O
only	O
sequences	O
.	O
</s>
<s>
The	O
canonical	O
LF	O
topology	O
makes	O
into	O
a	O
complete	B-Algorithm
distinguished	B-Algorithm
strict	B-Algorithm
LF-space	I-Algorithm
(	O
and	O
a	O
strict	B-Algorithm
LB-space	I-Algorithm
if	O
and	O
only	O
if	O
)	O
,	O
which	O
implies	O
that	O
is	O
a	O
meager	O
subset	O
of	O
itself	O
.	O
</s>
<s>
Furthermore	O
,	O
as	O
well	O
as	O
its	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
,	O
is	O
a	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
barrelled	B-Algorithm
bornological	B-Algorithm
Mackey	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
so	O
in	O
particular	O
,	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
which	O
is	O
the	O
space	O
of	O
distributions	O
on	O
,	O
is	O
metrizable	B-Algorithm
(	O
note	O
that	O
the	O
weak-*	O
topology	O
on	O
also	O
is	O
not	O
metrizable	B-Algorithm
and	O
moreover	O
,	O
it	O
further	O
lacks	O
almost	O
all	O
of	O
the	O
nice	O
properties	O
that	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
gives	O
)	O
.	O
</s>
<s>
The	O
three	O
spaces	O
and	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
as	O
well	O
as	O
the	O
strong	B-Algorithm
duals	I-Algorithm
of	O
each	O
of	O
these	O
three	O
spaces	O
,	O
are	O
complete	B-Algorithm
nuclear	B-Algorithm
Montel	B-Algorithm
bornological	B-Algorithm
spaces	I-Algorithm
,	O
which	O
implies	O
that	O
all	O
six	O
of	O
these	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
are	O
also	O
paracompact	O
reflexive	O
barrelled	B-Algorithm
Mackey	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
The	O
spaces	O
and	O
are	O
both	O
distinguished	B-Algorithm
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
both	O
and	O
are	O
Schwartz	B-Algorithm
TVSs	I-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
spaces	I-Algorithm
of	O
and	O
are	O
sequential	O
spaces	O
but	O
not	O
Fréchet-Urysohn	O
spaces	O
.	O
</s>
<s>
Moreover	O
,	O
neither	O
the	O
space	O
of	O
test	O
functions	O
nor	O
its	O
strong	B-Algorithm
dual	I-Algorithm
is	O
a	O
sequential	O
space	O
(	O
not	O
even	O
an	O
Ascoli	O
space	O
)	O
,	O
which	O
in	O
particular	O
implies	O
that	O
their	O
topologies	O
can	O
be	O
defined	O
entirely	O
in	O
terms	O
of	O
convergent	O
sequences	O
.	O
</s>
<s>
Neither	O
the	O
space	O
nor	O
its	O
strong	B-Algorithm
dual	I-Algorithm
is	O
a	O
sequential	O
space	O
,	O
and	O
consequently	O
,	O
their	O
topologies	O
can	O
be	O
defined	O
entirely	O
in	O
terms	O
of	O
convergent	O
sequences	O
.	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
which	O
in	O
particular	O
,	O
is	O
the	O
reason	O
why	O
a	O
sequence	O
of	O
distributions	O
converges	O
(	O
in	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
)	O
if	O
and	O
only	O
if	O
it	O
converges	O
pointwise	O
(	O
this	O
leads	O
many	O
authors	O
to	O
use	O
pointwise	O
convergence	O
to	O
actually	O
the	O
convergence	O
of	O
a	O
sequence	O
of	O
distributions	O
;	O
this	O
is	O
fine	O
for	O
sequences	O
but	O
it	O
does	O
extend	O
to	O
the	O
convergence	O
of	O
nets	O
of	O
distributions	O
since	O
a	O
net	O
may	O
converge	O
pointwise	O
but	O
fail	O
to	O
converge	O
in	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
)	O
.	O
</s>
<s>
Sequences	O
characterize	O
continuity	O
of	O
linear	O
maps	O
valued	O
in	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Suppose	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
(	O
such	O
as	O
any	O
of	O
the	O
six	O
TVSs	O
mentioned	O
earlier	O
)	O
.	O
</s>
<s>
Then	O
a	O
linear	O
map	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
maps	O
null	O
sequences	O
in	O
to	O
bounded	B-Algorithm
subsets	I-Algorithm
of	O
.	O
</s>
<s>
More	O
generally	O
,	O
such	O
a	O
linear	O
map	O
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
maps	O
Mackey	O
convergent	O
null	O
sequences	O
to	O
bounded	B-Algorithm
subsets	I-Algorithm
of	O
So	O
in	O
particular	O
,	O
if	O
a	O
linear	O
map	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
sequentially	O
continuous	B-Algorithm
at	O
the	O
origin	O
then	O
it	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
However	O
,	O
this	O
does	O
necessarily	O
extend	O
to	O
non-linear	O
maps	O
and/or	O
to	O
maps	O
valued	O
in	O
topological	O
spaces	O
that	O
are	O
not	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
is	O
TVS	B-Architecture
isomorphic	O
to	O
via	O
the	O
canonical	O
TVS-isomorphism	O
defined	O
by	O
sending	O
to	O
(	O
that	O
is	O
,	O
to	O
the	O
linear	B-Algorithm
functional	I-Algorithm
on	O
defined	O
by	O
sending	O
to	O
)	O
;	O
</s>
<s>
On	O
any	O
bounded	B-Algorithm
subset	O
of	O
the	O
weak	O
and	O
strong	O
subspace	O
topologies	O
coincide	O
;	O
the	O
same	O
is	O
true	O
for	O
;	O
</s>
<s>
Operations	O
on	O
distributions	O
and	O
spaces	O
of	O
distributions	O
are	O
often	O
defined	O
by	O
means	O
of	O
the	O
transpose	B-Algorithm
of	O
a	O
linear	O
operator	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
transpose	B-Algorithm
allows	O
for	O
a	O
unified	O
presentation	O
of	O
the	O
many	O
definitions	O
in	O
the	O
theory	O
of	O
distributions	O
and	O
also	O
because	O
its	O
properties	O
are	O
well	O
known	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
well-known	O
Hermitian	O
adjoint	O
of	O
a	O
linear	O
operator	O
between	O
Hilbert	O
spaces	O
is	O
just	O
the	O
operator	O
's	O
transpose	B-Algorithm
(	O
but	O
with	O
the	O
Riesz	O
representation	O
theorem	O
used	O
to	O
identify	O
each	O
Hilbert	O
space	O
with	O
its	O
continuous	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
or	O
equivalently	O
,	O
it	O
is	O
the	O
unique	O
map	O
satisfying	O
for	O
all	O
and	O
all	O
(	O
the	O
prime	O
symbol	O
in	O
does	O
not	O
denote	O
a	O
derivative	O
of	O
any	O
kind	O
;	O
it	O
merely	O
indicates	O
that	O
is	O
an	O
element	O
of	O
the	O
continuous	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
Since	O
is	O
continuous	B-Algorithm
,	O
the	O
transpose	B-Algorithm
is	O
also	O
continuous	B-Algorithm
when	O
both	O
duals	O
are	O
endowed	O
with	O
their	O
respective	O
strong	B-Algorithm
dual	I-Algorithm
topologies	I-Algorithm
;	O
it	O
is	O
also	O
continuous	B-Algorithm
when	O
both	O
duals	O
are	O
endowed	O
with	O
their	O
respective	O
weak*	O
topologies	O
(	O
see	O
the	O
articles	O
polar	O
topology	O
and	O
dual	B-Algorithm
system	I-Algorithm
for	O
more	O
details	O
)	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
distributions	O
,	O
the	O
characterization	O
of	O
the	O
transpose	B-Algorithm
can	O
be	O
refined	O
slightly	O
.	O
</s>
<s>
Let	O
be	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Then	O
by	O
definition	O
,	O
the	O
transpose	B-Algorithm
of	O
is	O
the	O
unique	O
linear	O
operator	O
that	O
satisfies	O
:	O
</s>
<s>
Since	O
is	O
dense	O
in	O
(	O
here	O
,	O
actually	O
refers	O
to	O
the	O
set	O
of	O
distributions	O
)	O
it	O
is	O
sufficient	O
that	O
the	O
defining	O
equality	O
hold	O
for	O
all	O
distributions	O
of	O
the	O
form	O
where	O
Explicitly	O
,	O
this	O
means	O
that	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
is	O
equal	O
to	O
if	O
and	O
only	O
if	O
the	O
condition	O
below	O
holds	O
:	O
</s>
<s>
which	O
is	O
a	O
continuous	B-Algorithm
injective	O
linear	O
map	O
.	O
</s>
<s>
If	O
then	O
the	O
(	O
continuous	B-Algorithm
injective	O
linear	O
)	O
trivial	O
extension	O
map	O
is	O
a	O
topological	O
embedding	O
(	O
in	O
other	O
words	O
,	O
if	O
this	O
linear	O
injection	O
was	O
used	O
to	O
identify	O
as	O
a	O
subset	O
of	O
then	O
'	O
s	O
topology	O
would	O
strictly	O
finer	O
than	O
the	O
subspace	O
topology	O
that	O
induces	O
on	O
it	O
;	O
importantly	O
,	O
it	O
would	O
be	O
a	O
topological	O
subspace	O
since	O
that	O
requires	O
equality	O
of	O
topologies	O
)	O
and	O
its	O
range	O
is	O
also	O
dense	O
in	O
its	O
codomain	B-Algorithm
Consequently	O
,	O
if	O
then	O
the	O
restriction	O
mapping	O
is	O
neither	O
injective	O
nor	O
surjective	B-Algorithm
.	O
</s>
<s>
Unless	O
the	O
restriction	O
to	O
is	O
neither	O
injective	O
nor	O
surjective	B-Algorithm
.	O
</s>
<s>
For	O
all	O
and	O
all	O
all	O
of	O
the	O
following	O
canonical	B-Algorithm
injections	I-Algorithm
are	O
continuous	B-Algorithm
and	O
have	O
an	O
image/range	O
that	O
is	O
a	O
dense	O
subset	O
of	O
their	O
codomain	B-Algorithm
:	O
</s>
<s>
where	O
the	O
topologies	O
on	O
the	O
LB-spaces	B-Algorithm
are	O
the	O
canonical	O
LF	O
topologies	O
as	O
defined	O
below	O
(	O
so	O
in	O
particular	O
,	O
they	O
are	O
not	O
the	O
usual	O
norm	O
topologies	O
)	O
.	O
</s>
<s>
The	O
range	O
of	O
each	O
of	O
the	O
maps	O
above	O
(	O
and	O
of	O
any	O
composition	O
of	O
the	O
maps	O
above	O
)	O
is	O
dense	O
in	O
the	O
codomain	B-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
is	O
even	O
sequentially	O
dense	O
in	O
every	O
For	O
every	O
the	O
canonical	O
inclusion	O
into	O
the	O
normed	O
space	O
(	O
here	O
has	O
its	O
usual	O
norm	O
topology	O
)	O
is	O
a	O
continuous	B-Algorithm
linear	O
injection	O
and	O
the	O
range	O
of	O
this	O
injection	O
is	O
dense	O
in	O
its	O
codomain	B-Algorithm
if	O
and	O
only	O
if	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
one	O
of	O
the	O
LF-spaces	B-Algorithm
(	O
for	O
)	O
or	O
LB-spaces	B-Algorithm
(	O
for	O
)	O
or	O
normed	O
spaces	O
(	O
for	O
)	O
.	O
</s>
<s>
Because	O
the	O
canonical	B-Algorithm
injection	I-Algorithm
is	O
a	O
continuous	B-Algorithm
injection	O
whose	O
image	O
is	O
dense	O
in	O
the	O
codomain	B-Algorithm
,	O
this	O
map	O
's	O
transpose	B-Algorithm
is	O
a	O
continuous	B-Algorithm
injection	O
.	O
</s>
<s>
This	O
injective	O
transpose	B-Algorithm
map	O
thus	O
allows	O
the	O
continuous	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
to	O
be	O
identified	O
with	O
a	O
certain	O
vector	O
subspace	O
of	O
the	O
space	O
of	O
all	O
distributions	O
(	O
specifically	O
,	O
it	O
is	O
identified	O
with	O
the	O
image	O
of	O
this	O
transpose	B-Algorithm
map	O
)	O
.	O
</s>
<s>
A	O
linear	O
subspace	O
of	O
carrying	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
that	O
is	O
finer	O
than	O
the	O
subspace	O
topology	O
induced	O
by	O
is	O
called	O
.	O
</s>
<s>
This	O
topology	O
makes	O
into	O
an	O
LB-space	B-Algorithm
(	O
and	O
thus	O
also	O
an	O
LF-space	B-Algorithm
)	O
with	O
a	O
topology	O
that	O
is	O
strictly	O
finer	O
than	O
the	O
norm	O
(	O
subspace	O
)	O
topology	O
that	O
induces	O
on	O
it	O
.	O
</s>
<s>
The	O
inclusion	B-Algorithm
map	I-Algorithm
is	O
a	O
continuous	B-Algorithm
injection	O
whose	O
image	O
is	O
dense	O
in	O
its	O
codomain	B-Algorithm
,	O
so	O
the	O
transpose	B-Algorithm
is	O
also	O
a	O
continuous	B-Algorithm
injection	O
.	O
</s>
<s>
Note	O
that	O
the	O
continuous	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
can	O
be	O
identified	O
as	O
the	O
space	O
of	O
Radon	O
measures	O
,	O
where	O
there	O
is	O
a	O
one-to-one	O
correspondence	O
between	O
the	O
continuous	B-Algorithm
linear	B-Algorithm
functionals	I-Algorithm
and	O
integral	O
with	O
respect	O
to	O
a	O
Radon	O
measure	O
;	O
that	O
is	O
,	O
</s>
<s>
if	O
is	O
a	O
Radon	O
measure	O
on	O
then	O
the	O
linear	B-Algorithm
functional	I-Algorithm
on	O
defined	O
by	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
A	O
linear	O
function	O
on	O
a	O
space	O
of	O
functions	O
is	O
called	O
if	O
whenever	O
a	O
function	O
that	O
belongs	O
to	O
the	O
domain	B-Algorithm
of	O
is	O
non-negative	O
(	O
meaning	O
that	O
is	O
real-valued	O
and	O
)	O
then	O
One	O
may	O
show	O
that	O
every	O
positive	O
linear	B-Algorithm
functional	I-Algorithm
on	O
is	O
necessarily	O
continuous	B-Algorithm
(	O
that	O
is	O
,	O
necessarily	O
a	O
Radon	O
measure	O
)	O
.	O
</s>
<s>
This	O
is	O
a	O
large	O
class	O
of	O
functions	O
which	O
includes	O
all	O
continuous	B-Algorithm
functions	O
and	O
all	O
Lp	O
space	O
functions	O
.	O
</s>
<s>
The	O
topology	O
on	O
is	O
defined	O
in	O
such	O
a	O
fashion	O
that	O
any	O
locally	O
integrable	O
function	O
yields	O
a	O
continuous	B-Algorithm
linear	B-Algorithm
functional	I-Algorithm
on	O
–	O
that	O
is	O
,	O
an	O
element	O
of	O
–	O
denoted	O
here	O
by	O
,	O
whose	O
value	O
on	O
the	O
test	O
function	O
is	O
given	O
by	O
the	O
Lebesgue	O
integral	O
:	O
</s>
<s>
In	O
a	O
similar	O
manner	O
,	O
every	O
Radon	O
measure	O
on	O
defines	O
an	O
element	O
of	O
whose	O
value	O
on	O
the	O
test	O
function	O
is	O
As	O
above	O
,	O
it	O
is	O
conventional	O
to	O
abuse	O
notation	O
and	O
write	O
the	O
pairing	B-Algorithm
between	O
a	O
Radon	O
measure	O
and	O
a	O
test	O
function	O
as	O
Conversely	O
,	O
as	O
shown	O
in	O
a	O
theorem	O
by	O
Schwartz	B-Algorithm
(	O
similar	O
to	O
the	O
Riesz	O
representation	O
theorem	O
)	O
,	O
every	O
distribution	O
which	O
is	O
non-negative	O
on	O
non-negative	O
functions	O
is	O
of	O
this	O
form	O
for	O
some	O
(	O
positive	O
)	O
Radon	O
measure	O
.	O
</s>
<s>
The	O
space	O
of	O
test	O
functions	O
is	O
sequentially	O
dense	O
in	O
with	O
respect	O
to	O
the	O
strong	B-Algorithm
topology	I-Algorithm
on	O
This	O
means	O
that	O
for	O
any	O
there	O
is	O
a	O
sequence	O
of	O
test	O
functions	O
,	O
that	O
converges	O
to	O
(	O
in	O
its	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
)	O
when	O
considered	O
as	O
a	O
sequence	O
of	O
distributions	O
.	O
</s>
<s>
The	O
inclusion	B-Algorithm
map	I-Algorithm
is	O
a	O
continuous	B-Algorithm
injection	O
whose	O
image	O
is	O
dense	O
in	O
its	O
codomain	B-Algorithm
,	O
so	O
the	O
transpose	B-Algorithm
is	O
also	O
a	O
continuous	B-Algorithm
injection	O
.	O
</s>
<s>
Thus	O
the	O
image	O
of	O
the	O
transpose	B-Algorithm
,	O
denoted	O
by	O
forms	O
a	O
space	O
of	O
distributions	O
when	O
it	O
is	O
endowed	O
with	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
of	O
(	O
transferred	O
to	O
it	O
via	O
the	O
transpose	B-Algorithm
map	O
so	O
the	O
topology	O
of	O
is	O
finer	O
than	O
the	O
subspace	O
topology	O
that	O
this	O
set	O
inherits	O
from	O
)	O
.	O
</s>
<s>
the	O
restriction	O
of	O
to	O
when	O
that	O
space	O
is	O
equipped	O
with	O
the	O
subspace	O
topology	O
inherited	O
from	O
(	O
a	O
coarser	O
topology	O
than	O
the	O
canonical	O
LF	O
topology	O
)	O
,	O
is	O
continuous	B-Algorithm
;	O
</s>
<s>
Compactly	O
supported	O
distributions	O
define	O
continuous	B-Algorithm
linear	B-Algorithm
functionals	I-Algorithm
on	O
the	O
space	O
;	O
recall	O
that	O
the	O
topology	O
on	O
is	O
defined	O
such	O
that	O
a	O
sequence	O
of	O
test	O
functions	O
converges	O
to	O
0	O
if	O
and	O
only	O
if	O
all	O
derivatives	O
of	O
converge	O
uniformly	B-Algorithm
to	O
0	O
on	O
every	O
compact	O
subset	O
of	O
.	O
</s>
<s>
Conversely	O
,	O
it	O
can	O
be	O
shown	O
that	O
every	O
continuous	B-Algorithm
linear	B-Algorithm
functional	I-Algorithm
on	O
this	O
space	O
defines	O
a	O
distribution	O
of	O
compact	O
support	O
.	O
</s>
<s>
Let	O
The	O
inclusion	B-Algorithm
map	I-Algorithm
is	O
a	O
continuous	B-Algorithm
injection	O
whose	O
image	O
is	O
dense	O
in	O
its	O
codomain	B-Algorithm
,	O
so	O
the	O
transpose	B-Algorithm
is	O
also	O
a	O
continuous	B-Algorithm
injection	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
image	O
of	O
denoted	O
by	O
forms	O
a	O
space	O
of	O
distributions	O
when	O
it	O
is	O
endowed	O
with	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
of	O
(	O
transferred	O
to	O
it	O
via	O
the	O
transpose	B-Algorithm
map	O
so	O
'	O
s	O
topology	O
is	O
finer	O
than	O
the	O
subspace	O
topology	O
that	O
this	O
set	O
inherits	O
from	O
)	O
.	O
</s>
<s>
The	O
Schwartz	B-Algorithm
space	I-Algorithm
,	O
is	O
the	O
space	O
of	O
all	O
smooth	O
functions	O
that	O
are	O
rapidly	O
decreasing	O
at	O
infinity	O
along	O
with	O
all	O
partial	O
derivatives	O
.	O
</s>
<s>
Thus	O
is	O
in	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
provided	O
that	O
any	O
derivative	O
of	O
multiplied	O
with	O
any	O
power	O
of	O
converges	O
to	O
0	O
as	O
These	O
functions	O
form	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
with	O
a	O
suitably	O
defined	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
Then	O
is	O
in	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
if	O
all	O
the	O
values	O
satisfy	O
:	O
</s>
<s>
The	O
family	O
of	O
seminorms	O
defines	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
For	O
the	O
seminorms	O
are	O
,	O
in	O
fact	O
,	O
norms	O
on	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
Schwartz	B-Algorithm
space	I-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
(	O
i.e.	O
</s>
<s>
a	O
complete	B-Algorithm
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
Because	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
changes	O
into	O
multiplication	O
by	O
and	O
vice	O
versa	O
,	O
this	O
symmetry	O
implies	O
that	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
Schwartz	B-Algorithm
function	I-Algorithm
is	O
also	O
a	O
Schwartz	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
is	O
dense	O
in	O
The	O
subset	O
of	O
all	O
analytic	O
Schwartz	B-Algorithm
functions	I-Algorithm
is	O
dense	O
in	O
as	O
well	O
.	O
</s>
<s>
where	O
represents	O
the	O
completion	O
of	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
(	O
which	O
in	O
this	O
case	O
is	O
the	O
identical	O
to	O
the	O
completion	O
of	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
)	O
.	O
</s>
<s>
The	O
inclusion	B-Algorithm
map	I-Algorithm
is	O
a	O
continuous	B-Algorithm
injection	O
whose	O
image	O
is	O
dense	O
in	O
its	O
codomain	B-Algorithm
,	O
so	O
the	O
transpose	B-Algorithm
is	O
also	O
a	O
continuous	B-Algorithm
injection	O
.	O
</s>
<s>
Thus	O
,	O
the	O
image	O
of	O
the	O
transpose	B-Algorithm
map	O
,	O
denoted	O
by	O
forms	O
a	O
space	O
of	O
distributions	O
when	O
it	O
is	O
endowed	O
with	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
of	O
(	O
transferred	O
to	O
it	O
via	O
the	O
transpose	B-Algorithm
map	O
so	O
the	O
topology	O
of	O
is	O
finer	O
than	O
the	O
subspace	O
topology	O
that	O
this	O
set	O
inherits	O
from	O
)	O
.	O
</s>
<s>
It	O
is	O
the	O
continuous	B-Algorithm
dual	O
of	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Tempered	O
distributions	O
generalize	O
the	O
bounded	B-Algorithm
(	O
or	O
slow-growing	O
)	O
locally	O
integrable	O
functions	O
;	O
all	O
distributions	O
with	O
compact	O
support	O
and	O
all	O
square-integrable	B-Algorithm
functions	I-Algorithm
are	O
tempered	O
distributions	O
.	O
</s>
<s>
To	O
study	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
it	O
is	O
best	O
to	O
consider	O
complex-valued	O
test	O
functions	O
and	O
complex-linear	O
distributions	O
.	O
</s>
<s>
The	O
ordinary	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
a	O
TVS-automorphism	O
of	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
,	O
and	O
the	O
is	O
defined	O
to	O
be	O
its	O
transpose	B-Algorithm
which	O
(	O
abusing	O
notation	O
)	O
will	O
again	O
be	O
denoted	O
by	O
.	O
</s>
<s>
So	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
tempered	O
distribution	O
is	O
defined	O
by	O
for	O
every	O
Schwartz	B-Algorithm
function	I-Algorithm
is	O
thus	O
again	O
a	O
tempered	O
distribution	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
a	O
TVS	B-Architecture
isomorphism	O
from	O
the	O
space	O
of	O
tempered	O
distributions	O
onto	O
itself	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
constant	O
function	O
equal	O
to	O
1	O
is	O
the	O
distribution	O
.	O
</s>
<s>
These	O
definitions	O
associate	O
every	O
and	O
with	O
the	O
(	O
respective	O
)	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
:	O
</s>
<s>
has	O
compact	O
support	O
then	O
it	O
also	O
induces	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
of	O
(	O
resp	O
.	O
</s>
<s>
the	O
span	O
of	O
the	O
range	O
of	O
this	O
map	O
is	O
a	O
dense	O
subspace	O
of	O
its	O
codomain	B-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
Moreover	O
induces	O
continuous	B-Algorithm
bilinear	O
maps	O
:	O
</s>
<s>
where	O
denotes	O
the	O
space	O
of	O
distributions	O
with	O
compact	O
support	O
and	O
is	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
of	O
rapidly	O
decreasing	O
functions	O
.	O
</s>
<s>
This	O
question	O
led	O
Alexander	O
Grothendieck	O
to	O
discover	O
nuclear	B-Algorithm
spaces	I-Algorithm
,	O
nuclear	B-Algorithm
maps	I-Algorithm
,	O
and	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
.	O
</s>
<s>
He	O
ultimately	O
showed	O
that	O
it	O
is	O
precisely	O
because	O
is	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
that	O
the	O
Schwartz	B-Algorithm
kernel	O
theorem	O
holds	O
.	O
</s>
<s>
Like	O
Hilbert	O
spaces	O
,	O
nuclear	B-Algorithm
spaces	I-Algorithm
may	O
be	O
thought	O
as	O
of	O
generalizations	O
of	O
finite	O
dimensional	O
Euclidean	O
space	O
.	O
</s>
<s>
A	O
refined	O
theory	O
has	O
been	O
developed	O
,	O
in	O
particular	O
Mikio	O
Sato	O
's	O
algebraic	O
analysis	O
,	O
using	O
sheaf	O
theory	O
and	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
