<s>
An	O
integral	O
bilinear	O
form	O
is	O
a	O
bilinear	O
functional	O
that	O
belongs	O
to	O
the	O
continuous	O
dual	O
space	O
of	O
,	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
the	O
locally	O
convex	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
X	O
and	O
Y	O
.	O
</s>
<s>
An	O
integral	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
is	O
a	O
continuous	O
linear	O
operator	O
that	O
arises	O
in	O
a	O
canonical	O
way	O
from	O
an	O
integral	O
bilinear	O
form	O
.	O
</s>
<s>
These	O
maps	O
play	O
an	O
important	O
role	O
in	O
the	O
theory	O
of	O
nuclear	B-Algorithm
spaces	I-Algorithm
and	O
nuclear	B-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
Let	O
X	O
and	O
Y	O
be	O
locally	O
convex	O
TVSs	O
,	O
let	O
denote	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
,	O
denote	O
its	O
completion	O
,	O
let	O
denote	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
,	O
and	O
denote	O
its	O
completion	O
.	O
</s>
<s>
It	O
follows	O
that	O
an	O
integral	B-Algorithm
map	I-Algorithm
is	O
of	O
the	O
form	O
:	O
</s>
<s>
The	O
above	O
integral	O
is	O
the	O
weak	B-Algorithm
integral	I-Algorithm
,	O
so	O
the	O
equality	O
holds	O
if	O
and	O
only	O
if	O
for	O
every	O
,	O
.	O
</s>
<s>
An	O
integral	B-Algorithm
map	I-Algorithm
is	O
of	O
the	O
form	O
,	O
for	O
every	O
and	O
:	O
</s>
<s>
The	O
following	O
result	O
shows	O
that	O
integral	B-Algorithm
maps	I-Algorithm
"	O
factor	O
through	O
"	O
Hilbert	O
spaces	O
.	O
</s>
<s>
Proposition	O
:	O
Suppose	O
that	O
is	O
an	O
integral	B-Algorithm
map	I-Algorithm
between	O
locally	O
convex	O
TVS	O
with	O
Y	O
Hausdorff	O
and	O
complete	O
.	O
</s>
<s>
Furthermore	O
,	O
every	O
integral	O
operator	O
between	O
two	O
Hilbert	O
spaces	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Thus	O
a	O
continuous	O
linear	O
operator	O
between	O
two	O
Hilbert	O
spaces	O
is	O
nuclear	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
integral	O
.	O
</s>
<s>
Every	O
nuclear	B-Algorithm
map	I-Algorithm
is	O
integral	O
.	O
</s>
<s>
An	O
important	O
partial	O
converse	O
is	O
that	O
every	O
integral	O
operator	O
between	O
two	O
Hilbert	O
spaces	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
If	O
,	O
,	O
and	O
are	O
all	O
integral	O
linear	O
maps	O
then	O
their	O
composition	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Thus	O
,	O
in	O
particular	O
,	O
if	O
is	O
an	O
infinite-dimensional	O
Fréchet	B-Algorithm
space	I-Algorithm
then	O
a	O
continuous	O
linear	O
surjection	O
cannot	O
be	O
an	O
integral	O
operator	O
.	O
</s>
