<s>
In	O
mathematics	O
,	O
nuclear	B-Algorithm
operators	I-Algorithm
are	O
an	O
important	O
class	O
of	O
linear	O
operators	O
introduced	O
by	O
Alexander	O
Grothendieck	O
in	O
his	O
doctoral	O
dissertation	O
.	O
</s>
<s>
Nuclear	B-Algorithm
operators	I-Algorithm
are	O
intimately	O
tied	O
to	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
two	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
.	O
</s>
<s>
Throughout	O
let	O
X	O
,	O
Y	O
,	O
and	O
Z	O
be	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
and	O
L	O
:	O
X	O
→	O
Y	O
be	O
a	O
linear	O
operator	O
(	O
no	O
assumption	O
of	O
continuity	O
is	O
made	O
unless	O
otherwise	O
stated	O
)	O
.	O
</s>
<s>
The	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
two	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
X	O
and	O
Y	O
is	O
denoted	O
by	O
and	O
the	O
completion	O
of	O
this	O
space	O
will	O
be	O
denoted	O
by	O
.	O
</s>
<s>
L	O
:	O
X	O
→	O
Y	O
is	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
or	O
homomorphism	O
,	O
if	O
it	O
is	O
linear	O
,	O
continuous	O
,	O
and	O
is	O
an	O
open	O
map	O
,	O
where	O
,	O
the	O
image	O
of	O
L	O
,	O
has	O
the	O
subspace	O
topology	O
induced	O
by	O
Y	O
.	O
</s>
<s>
As	O
usual	O
,	O
if	O
X*	O
is	O
considered	O
as	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
but	O
it	O
has	O
not	O
been	O
made	O
clear	O
what	O
topology	O
it	O
is	O
endowed	O
with	O
,	O
then	O
the	O
topology	O
will	O
be	O
assumed	O
to	O
be	O
b( X′	O
,	O
X	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
if	O
X	O
and	O
Y	O
are	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
(	O
TVSs	O
)	O
and	O
if	O
X	O
⊗	O
Y	O
is	O
given	O
the	O
𝜋-topology	O
then	O
for	O
every	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
Z	O
,	O
this	O
map	O
restricts	O
to	O
a	O
vector	O
space	O
isomorphism	O
from	O
the	O
space	O
of	O
continuous	O
linear	O
mappings	O
onto	O
the	O
space	O
of	O
continuous	O
bilinear	O
mappings	O
.	O
</s>
<s>
The	O
range	O
of	O
this	O
map	O
is	O
denoted	O
by	O
and	O
its	O
elements	O
are	O
called	O
nuclear	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
Explicitly	O
,	O
if	O
is	O
a	O
nuclear	B-Algorithm
operator	I-Algorithm
then	O
.	O
</s>
<s>
If	O
is	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
then	O
its	O
transpose	O
is	O
a	O
continuous	O
nuclear	B-Algorithm
map	I-Algorithm
(	O
when	O
the	O
dual	O
spaces	O
carry	O
their	O
strong	O
dual	O
topologies	O
)	O
and	O
.	O
</s>
<s>
Then	O
N	O
is	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
if	O
and	O
only	O
if	O
R	O
is	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
;	O
</s>
<s>
hence	O
,	O
to	O
study	O
nuclear	B-Algorithm
maps	I-Algorithm
between	O
Hilbert	O
spaces	O
it	O
suffices	O
to	O
restrict	O
one	O
's	O
attention	O
to	O
positive	O
self-adjoint	O
operators	O
.	O
</s>
<s>
N	O
:	O
X	O
→	O
Y	O
is	O
an	O
integral	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
U	O
is	O
a	O
convex	O
balanced	O
closed	O
neighborhood	O
of	O
the	O
origin	O
in	O
X	O
and	O
B	O
is	O
a	O
convex	O
balanced	O
bounded	O
Banach	B-Algorithm
disk	I-Algorithm
in	O
Y	O
with	O
both	O
X	O
and	O
Y	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
One	O
can	O
define	O
the	O
auxiliary	B-Algorithm
Banach	I-Algorithm
space	I-Algorithm
with	O
the	O
canonical	O
map	O
whose	O
image	O
,	O
,	O
is	O
dense	O
in	O
as	O
well	O
as	O
the	O
auxiliary	O
space	O
normed	O
by	O
and	O
with	O
a	O
canonical	O
map	O
being	O
the	O
(	O
continuous	O
)	O
canonical	O
injection	O
.	O
</s>
<s>
Definition	O
:	O
Let	O
X	O
and	O
Y	O
be	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
The	O
union	O
of	O
all	O
as	O
U	O
ranges	O
over	O
all	O
closed	O
convex	O
balanced	O
neighborhoods	O
of	O
the	O
origin	O
in	O
X	O
and	O
B	O
ranges	O
over	O
all	O
bounded	O
Banach	B-Algorithm
disks	I-Algorithm
in	O
Y	O
,	O
is	O
denoted	O
by	O
and	O
its	O
elements	O
are	O
call	O
nuclear	O
mappings	O
of	O
X	O
into	O
Y	O
.	O
</s>
<s>
Let	O
W	O
,	O
X	O
,	O
Y	O
,	O
and	O
Z	O
be	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
,	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
,	O
and	O
and	O
be	O
continuous	O
linear	O
maps	O
.	O
</s>
<s>
If	O
is	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
between	O
two	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
,	O
then	O
its	O
transpose	O
is	O
a	O
continuous	O
nuclear	B-Algorithm
map	I-Algorithm
(	O
when	O
the	O
dual	O
spaces	O
carry	O
their	O
strong	O
dual	O
topologies	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
between	O
two	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
and	O
if	O
is	O
a	O
completion	O
of	O
X	O
,	O
then	O
the	O
unique	O
continuous	O
extension	O
of	O
N	O
is	O
nuclear	O
.	O
</s>
<s>
Let	O
X	O
and	O
Y	O
be	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
and	O
let	O
be	O
a	O
continuous	O
linear	O
operator	O
.	O
</s>
<s>
(	O
Definition	O
)	O
There	O
exists	O
a	O
convex	O
balanced	O
neighborhood	O
U	O
of	O
the	O
origin	O
in	O
X	O
and	O
a	O
bounded	O
Banach	B-Algorithm
disk	I-Algorithm
B	O
in	O
Y	O
such	O
that	O
and	O
the	O
induced	O
map	O
is	O
nuclear	O
,	O
where	O
is	O
the	O
unique	O
continuous	O
extension	O
of	O
,	O
which	O
is	O
the	O
unique	O
map	O
satisfying	O
where	O
is	O
the	O
natural	O
inclusion	O
and	O
is	O
the	O
canonical	O
projection	O
.	O
</s>
<s>
There	O
exists	O
an	O
equicontinuous	O
sequence	O
in	O
,	O
a	O
bounded	O
Banach	B-Algorithm
disk	I-Algorithm
,	O
a	O
sequence	O
in	O
B	O
,	O
and	O
a	O
complex	O
sequence	O
such	O
that	O
and	O
is	O
equal	O
to	O
the	O
mapping	O
:	O
for	O
all	O
.	O
</s>
<s>
If	O
X	O
is	O
barreled	O
and	O
Y	O
is	O
quasi-complete	B-Algorithm
,	O
then	O
N	O
is	O
nuclear	O
if	O
and	O
only	O
if	O
N	O
has	O
a	O
representation	O
of	O
the	O
form	O
with	O
bounded	O
in	O
,	O
bounded	O
in	O
Y	O
and	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
type	O
of	O
Hahn-Banach	O
theorem	O
for	O
extending	O
nuclear	B-Algorithm
maps	I-Algorithm
:	O
</s>
<s>
If	O
is	O
a	O
TVS-embedding	O
and	O
is	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
then	O
there	O
exists	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
such	O
that	O
.	O
</s>
<s>
Suppose	O
all	O
that	O
every	O
compact	O
disk	O
in	O
is	O
the	O
image	O
under	O
of	O
a	O
bounded	O
Banach	B-Algorithm
disk	I-Algorithm
in	O
Z	O
(	O
this	O
is	O
true	O
,	O
for	O
instance	O
,	O
if	O
X	O
and	O
Z	O
are	O
both	O
Fréchet	O
spaces	O
,	O
or	O
if	O
Z	O
is	O
the	O
strong	O
dual	O
of	O
a	O
Fréchet	O
space	O
and	O
is	O
weakly	O
closed	O
in	O
Z	O
)	O
.	O
</s>
<s>
Then	O
for	O
every	O
nuclear	B-Algorithm
map	I-Algorithm
there	O
exists	O
a	O
nuclear	B-Algorithm
map	I-Algorithm
such	O
that	O
.	O
</s>
<s>
Let	O
X	O
and	O
Y	O
be	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
and	O
let	O
be	O
a	O
continuous	O
linear	O
operator	O
.	O
</s>
<s>
Any	O
nuclear	B-Algorithm
map	I-Algorithm
is	O
compact	O
.	O
</s>
<s>
For	O
every	O
topology	O
of	O
uniform	O
convergence	O
on	O
,	O
the	O
nuclear	B-Algorithm
maps	I-Algorithm
are	O
contained	O
in	O
the	O
closure	O
of	O
(	O
when	O
is	O
viewed	O
as	O
a	O
subspace	O
of	O
)	O
.	O
</s>
