<s>
In	O
mathematics	O
,	O
nuclear	B-Algorithm
spaces	I-Algorithm
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
that	O
can	O
be	O
viewed	O
as	O
a	O
generalization	O
of	O
finite	O
dimensional	O
Euclidean	O
spaces	O
and	O
share	O
many	O
of	O
their	O
desirable	O
properties	O
.	O
</s>
<s>
Nuclear	B-Algorithm
spaces	I-Algorithm
are	O
however	O
quite	O
different	O
from	O
Hilbert	O
spaces	O
,	O
another	O
generalization	O
of	O
finite	O
dimensional	O
Euclidean	O
spaces	O
.	O
</s>
<s>
The	O
topology	O
on	O
nuclear	B-Algorithm
spaces	I-Algorithm
can	O
be	O
defined	O
by	O
a	O
family	O
of	O
seminorms	O
whose	O
unit	O
balls	O
decrease	O
rapidly	O
in	O
size	O
.	O
</s>
<s>
Vector	O
spaces	O
whose	O
elements	O
are	O
"	O
smooth	O
"	O
in	O
some	O
sense	O
tend	O
to	O
be	O
nuclear	B-Algorithm
spaces	I-Algorithm
;	O
a	O
typical	O
example	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
the	O
set	O
of	O
smooth	O
functions	O
on	O
a	O
compact	O
manifold	O
.	O
</s>
<s>
All	O
finite-dimensional	O
vector	O
spaces	O
are	O
nuclear	B-Algorithm
.	O
</s>
<s>
There	O
are	O
no	O
Banach	O
spaces	O
that	O
are	O
nuclear	B-Algorithm
,	O
except	O
for	O
the	O
finite-dimensional	O
ones	O
.	O
</s>
<s>
In	O
practice	O
a	O
sort	O
of	O
converse	O
to	O
this	O
is	O
often	O
true	O
:	O
if	O
a	O
"	O
naturally	O
occurring	O
"	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
a	O
Banach	O
space	O
,	O
then	O
there	O
is	O
a	O
good	O
chance	O
that	O
it	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Much	O
of	O
the	O
theory	O
of	O
nuclear	B-Algorithm
spaces	I-Algorithm
was	O
developed	O
by	O
Alexander	O
Grothendieck	O
while	O
investigating	O
the	O
Schwartz	O
kernel	O
theorem	O
and	O
published	O
in	O
.	O
</s>
<s>
For	O
any	O
open	O
subsets	O
and	O
the	O
canonical	O
map	O
is	O
an	O
isomorphism	O
of	O
TVSs	O
(	O
where	O
has	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	I-Algorithm
bounded	I-Algorithm
subsets	I-Algorithm
)	O
and	O
furthermore	O
,	O
both	O
of	O
these	O
spaces	O
are	O
canonically	O
TVS-isomorphic	O
to	O
(	O
where	O
since	O
is	O
nuclear	B-Algorithm
,	O
this	O
tensor	O
product	O
is	O
simultaneously	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
and	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
)	O
.	O
</s>
<s>
This	O
question	O
led	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>
This	O
section	O
lists	O
some	O
of	O
the	O
more	O
common	O
definitions	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
some	O
authors	O
use	O
a	O
more	O
restrictive	O
definition	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
,	O
by	O
adding	O
the	O
condition	O
that	O
the	O
space	O
should	O
also	O
be	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
following	O
definition	O
was	O
used	O
by	O
Grothendieck	O
to	O
define	O
nuclear	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Definition	O
0	O
:	O
Let	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Then	O
is	O
nuclear	B-Algorithm
if	O
for	O
any	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
the	O
canonical	O
vector	O
space	O
embedding	O
is	O
an	O
embedding	O
of	O
TVSs	O
whose	O
image	O
is	O
dense	O
in	O
the	O
codomain	O
(	O
where	O
the	O
domain	O
is	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
and	O
the	O
codomain	O
is	O
the	O
space	O
of	O
all	O
separately	O
continuous	O
bilinear	O
forms	O
on	O
endowed	O
with	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	I-Algorithm
equicontinuous	I-Algorithm
subsets	I-Algorithm
)	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
has	O
a	O
topology	O
that	O
is	O
defined	O
by	O
some	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
If	O
is	O
a	O
seminorm	O
on	O
then	O
denotes	O
the	O
Banach	O
space	O
given	O
by	O
completing	B-Algorithm
the	O
auxiliary	B-Algorithm
normed	I-Algorithm
space	I-Algorithm
using	O
the	O
seminorm	O
There	O
is	O
a	O
natural	O
map	O
(	O
not	O
necessarily	O
injective	O
)	O
.	O
</s>
<s>
The	O
space	O
is	O
nuclear	B-Algorithm
when	O
a	O
stronger	O
condition	O
holds	O
,	O
namely	O
that	O
these	O
maps	O
are	O
nuclear	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
The	O
condition	O
of	O
being	O
a	O
nuclear	B-Algorithm
operator	I-Algorithm
is	O
subtle	O
,	O
and	O
more	O
details	O
are	O
available	O
in	O
the	O
corresponding	O
article	O
.	O
</s>
<s>
Definition	O
1	O
:	O
A	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
such	O
that	O
for	O
any	O
seminorm	O
we	O
can	O
find	O
a	O
larger	O
seminorm	O
so	O
that	O
the	O
natural	O
map	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Instead	O
of	O
using	O
arbitrary	O
Banach	O
spaces	O
and	O
nuclear	B-Algorithm
operators	I-Algorithm
,	O
we	O
can	O
give	O
a	O
definition	O
in	O
terms	O
of	O
Hilbert	O
spaces	O
and	O
trace	O
class	O
operators	O
,	O
which	O
are	O
easier	O
to	O
understand	O
.	O
</s>
<s>
(	O
On	O
Hilbert	O
spaces	O
nuclear	B-Algorithm
operators	I-Algorithm
are	O
often	O
called	O
trace	O
class	O
operators	O
.	O
)	O
</s>
<s>
Definition	O
2	O
:	O
A	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
with	O
a	O
topology	O
defined	O
by	O
a	O
family	O
of	O
Hilbert	O
seminorms	O
,	O
such	O
that	O
for	O
any	O
Hilbert	O
seminorm	O
we	O
can	O
find	O
a	O
larger	O
Hilbert	O
seminorm	O
so	O
that	O
the	O
natural	O
map	O
from	O
to	O
is	O
trace	O
class	O
.	O
</s>
<s>
Definition	O
3	O
:	O
A	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
with	O
a	O
topology	O
defined	O
by	O
a	O
family	O
of	O
Hilbert	O
seminorms	O
,	O
such	O
that	O
for	O
any	O
Hilbert	O
seminorm	O
we	O
can	O
find	O
a	O
larger	O
Hilbert	O
seminorm	O
so	O
that	O
the	O
natural	O
map	O
from	O
to	O
is	O
Hilbert	O
–	O
Schmidt	O
.	O
</s>
<s>
If	O
we	O
are	O
willing	O
to	O
use	O
the	O
concept	O
of	O
a	O
nuclear	B-Algorithm
operator	I-Algorithm
from	O
an	O
arbitrary	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
to	O
a	O
Banach	O
space	O
,	O
we	O
can	O
give	O
shorter	O
definitions	O
as	O
follows	O
:	O
</s>
<s>
Definition	O
4	O
:	O
A	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
such	O
that	O
for	O
any	O
seminorm	O
the	O
natural	O
map	O
from	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Definition	O
5	O
:	O
A	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
such	O
that	O
any	O
continuous	O
linear	O
map	O
to	O
a	O
Banach	O
space	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Definition	O
6	O
:	O
A	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
such	O
that	O
for	O
any	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
the	O
natural	O
map	O
from	O
the	O
projective	O
to	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
and	O
is	O
an	O
isomorphism	O
.	O
</s>
<s>
Let	O
be	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
is	O
nuclear	B-Algorithm
;	O
</s>
<s>
for	O
any	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
the	O
canonical	O
vector	O
space	O
embedding	O
is	O
an	O
embedding	O
of	O
TVSs	O
whose	O
image	O
is	O
dense	O
in	O
the	O
codomain	O
;	O
</s>
<s>
for	O
any	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
space	O
the	O
canonical	O
vector	O
space	O
embedding	O
is	O
a	O
surjective	O
isomorphism	O
of	O
TVSs	O
;	O
</s>
<s>
for	O
any	O
seminorm	O
we	O
can	O
find	O
a	O
larger	O
seminorm	O
so	O
that	O
the	O
natural	O
map	O
is	O
nuclear	B-Algorithm
;	O
</s>
<s>
for	O
any	O
seminorm	O
we	O
can	O
find	O
a	O
larger	O
seminorm	O
so	O
that	O
the	O
canonical	O
injection	O
is	O
nuclear	B-Algorithm
;	O
</s>
<s>
for	O
any	O
seminorm	O
the	O
natural	O
map	O
from	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
any	O
continuous	O
linear	O
map	O
to	O
a	O
Banach	O
space	O
is	O
nuclear	B-Algorithm
;	O
</s>
<s>
every	O
linear	O
map	O
from	O
a	O
Banach	O
space	O
into	O
that	O
transforms	O
the	O
unit	O
ball	O
into	O
an	O
equicontinuous	O
set	O
,	O
is	O
nuclear	B-Algorithm
;	O
</s>
<s>
the	O
completion	O
of	O
is	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
;	O
</s>
<s>
If	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
is	O
nuclear	B-Algorithm
;	O
</s>
<s>
the	O
strong	O
dual	O
of	O
is	O
nuclear	B-Algorithm
;	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
space	O
is	O
nuclear	B-Algorithm
if	O
and	O
only	O
if	O
its	O
completion	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Every	O
subspace	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Every	O
Hausdorff	O
quotient	O
space	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
inductive	O
limit	O
of	O
a	O
countable	O
sequence	O
of	O
nuclear	B-Algorithm
spaces	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
locally	B-Algorithm
convex	I-Algorithm
direct	O
sum	O
of	O
a	O
countable	O
sequence	O
of	O
nuclear	B-Algorithm
spaces	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
strong	O
dual	O
of	O
a	O
nuclear	B-Algorithm
Fréchet	B-Algorithm
space	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
the	O
strong	O
dual	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
may	O
fail	O
to	O
be	O
nuclear	B-Algorithm
.	O
</s>
<s>
A	O
Fréchet	B-Algorithm
space	I-Algorithm
whose	O
strong	O
dual	O
is	O
nuclear	B-Algorithm
is	O
itself	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
limit	O
of	O
a	O
family	O
of	O
nuclear	B-Algorithm
spaces	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
product	O
of	O
a	O
family	O
of	O
nuclear	B-Algorithm
spaces	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
completion	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
nuclear	B-Algorithm
(	O
and	O
in	O
fact	O
a	O
space	O
is	O
nuclear	B-Algorithm
if	O
and	O
only	O
if	O
its	O
completion	O
is	O
nuclear	B-Algorithm
)	O
.	O
</s>
<s>
The	O
tensor	O
product	O
of	O
two	O
nuclear	B-Algorithm
spaces	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
,	O
as	O
well	O
as	O
its	O
completion	O
,	O
of	O
two	O
nuclear	B-Algorithm
spaces	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
with	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
If	O
is	O
nuclear	B-Algorithm
then	O
the	O
vector	O
space	O
of	O
continuous	O
linear	O
maps	O
endowed	O
with	O
the	O
topology	O
of	O
simple	O
convergence	O
is	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
semi-reflexive	O
space	O
whose	O
strong	O
dual	O
is	O
nuclear	B-Algorithm
and	O
if	O
is	O
nuclear	B-Algorithm
then	O
the	O
vector	O
space	O
of	O
continuous	O
linear	O
maps	O
(	O
endowed	O
with	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	I-Algorithm
bounded	I-Algorithm
subsets	I-Algorithm
of	O
)	O
is	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
set	O
of	O
any	O
cardinality	O
,	O
then	O
and	O
(	O
with	O
the	O
product	O
topology	O
)	O
are	O
both	O
nuclear	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
A	O
relatively	O
simple	O
infinite	O
dimensional	O
example	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
the	O
space	O
of	O
all	O
rapidly	O
decreasing	O
sequences	O
(	O
"	O
Rapidly	O
decreasing	O
"	O
means	O
that	O
is	O
bounded	B-Algorithm
for	O
any	O
polynomial	O
)	O
.	O
</s>
<s>
If	O
the	O
completion	O
in	O
this	O
norm	O
is	O
then	O
there	O
is	O
a	O
natural	O
map	O
from	O
whenever	O
and	O
this	O
is	O
nuclear	B-Algorithm
whenever	O
essentially	O
because	O
the	O
series	O
is	O
then	O
absolutely	O
convergent	O
.	O
</s>
<s>
In	O
particular	O
for	O
each	O
norm	O
this	O
is	O
possible	O
to	O
find	O
another	O
norm	O
,	O
say	O
such	O
that	O
the	O
map	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
So	O
the	O
space	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
space	O
of	O
smooth	O
functions	O
on	O
any	O
compact	O
manifold	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
Schwartz	B-Algorithm
space	I-Algorithm
of	O
smooth	O
functions	O
on	O
for	O
which	O
the	O
derivatives	O
of	O
all	O
orders	O
are	O
rapidly	O
decreasing	O
is	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
space	O
of	O
entire	O
holomorphic	O
functions	O
on	O
the	O
complex	O
plane	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
space	O
of	O
distributions	O
the	O
strong	O
dual	O
of	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Nuclear	B-Algorithm
spaces	I-Algorithm
are	O
in	O
many	O
ways	O
similar	O
to	O
finite-dimensional	O
spaces	O
and	O
have	O
many	O
of	O
their	O
good	O
properties	O
.	O
</s>
<s>
Every	O
finite-dimensional	O
Hausdorff	O
space	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
A	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
nuclear	B-Algorithm
if	O
and	O
only	O
if	O
its	O
strong	O
dual	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Every	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
precompact	O
(	O
recall	O
that	O
a	O
set	O
is	O
precompact	O
if	O
its	O
closure	O
in	O
the	O
completion	O
of	O
the	O
space	O
is	O
compact	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
quasi-complete	B-Algorithm
(	O
i.e.	O
</s>
<s>
all	O
closed	O
and	O
bounded	B-Algorithm
subsets	O
are	O
complete	O
)	O
nuclear	B-Algorithm
space	I-Algorithm
then	O
has	O
the	O
Heine-Borel	O
property	O
.	O
</s>
<s>
A	O
nuclear	B-Algorithm
quasi-complete	B-Algorithm
barrelled	B-Algorithm
space	I-Algorithm
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
closed	O
equicontinuous	O
subset	O
of	O
the	O
dual	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
compact	O
metrizable	O
set	O
(	O
for	O
the	O
strong	O
dual	O
topology	O
)	O
.	O
</s>
<s>
Every	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
subspace	O
of	O
a	O
product	O
of	O
Hilbert	O
spaces	O
.	O
</s>
<s>
Every	O
nuclear	B-Algorithm
space	I-Algorithm
admits	O
a	O
basis	O
of	O
seminorms	O
consisting	O
of	O
Hilbert	O
norms	O
.	O
</s>
<s>
Every	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
Schwartz	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
nuclear	B-Algorithm
space	I-Algorithm
possesses	O
the	O
approximation	O
property	O
.	O
</s>
<s>
Any	O
subspace	O
and	O
any	O
quotient	O
space	O
by	O
a	O
closed	O
subspace	O
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
If	O
is	O
nuclear	B-Algorithm
and	O
is	O
any	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
,	O
then	O
the	O
natural	O
map	O
from	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
A	O
and	O
to	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
is	O
an	O
isomorphism	O
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
measures	O
on	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
a	O
basic	O
theorem	O
states	O
that	O
any	O
continuous	O
cylinder	B-Algorithm
set	I-Algorithm
measure	I-Algorithm
on	O
the	O
dual	O
of	O
a	O
nuclear	B-Algorithm
Fréchet	B-Algorithm
space	I-Algorithm
automatically	O
extends	O
to	O
a	O
Radon	O
measure	O
.	O
</s>
<s>
This	O
is	O
useful	O
because	O
it	O
is	O
often	O
easy	O
to	O
construct	O
cylinder	B-Algorithm
set	I-Algorithm
measures	I-Algorithm
on	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
but	O
these	O
are	O
not	O
good	O
enough	O
for	O
most	O
applications	O
unless	O
they	O
are	O
Radon	O
measures	O
(	O
for	O
example	O
,	O
they	O
are	O
not	O
even	O
countably	O
additive	O
in	O
general	O
)	O
.	O
</s>
<s>
Much	O
of	O
the	O
theory	O
of	O
nuclear	B-Algorithm
spaces	I-Algorithm
was	O
developed	O
by	O
Alexander	O
Grothendieck	O
while	O
investigating	O
the	O
Schwartz	O
kernel	O
theorem	O
and	O
published	O
in	O
.	O
</s>
<s>
Schwartz	O
kernel	O
theorem	O
:	O
Suppose	O
that	O
is	O
nuclear	B-Algorithm
,	O
is	O
locally	B-Algorithm
convex	I-Algorithm
,	O
and	O
is	O
a	O
continuous	O
bilinear	O
form	O
on	O
Then	O
originates	O
from	O
a	O
space	O
of	O
the	O
form	O
where	O
and	O
are	O
suitable	O
equicontinuous	O
subsets	O
of	O
and	O
Equivalently	O
,	O
is	O
of	O
the	O
form	O
,	O
</s>
<s>
This	O
extends	O
the	O
inverse	O
Fourier	O
transform	O
to	O
nuclear	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Such	O
measure	O
is	O
called	O
white	B-Algorithm
noise	I-Algorithm
measure	I-Algorithm
.	O
</s>
<s>
When	O
is	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
,	O
the	O
corresponding	O
random	O
element	O
is	O
a	O
random	O
distribution	O
.	O
</s>
<s>
A	O
strongly	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
such	O
that	O
for	O
any	O
seminorm	O
there	O
exists	O
a	O
larger	O
seminorm	O
so	O
that	O
the	O
natural	O
map	O
is	O
a	O
strongly	O
nuclear	B-Algorithm
.	O
</s>
