<s>
In	O
functional	B-Application
analysis	I-Application
,	O
the	O
Borel	B-Algorithm
graph	I-Algorithm
theorem	I-Algorithm
is	O
generalization	O
of	O
the	O
closed	O
graph	O
theorem	O
that	O
was	O
proven	O
by	O
L	O
.	O
Schwartz	O
.	O
</s>
<s>
The	O
Borel	B-Algorithm
graph	I-Algorithm
theorem	I-Algorithm
shows	O
that	O
the	O
closed	O
graph	O
theorem	O
is	O
valid	O
for	O
linear	B-Architecture
maps	I-Architecture
defined	O
on	O
and	O
valued	O
in	O
most	O
spaces	O
encountered	O
in	O
analysis	O
.	O
</s>
<s>
The	O
weak	O
dual	O
of	O
a	O
separable	O
Fréchet	B-Algorithm
space	I-Algorithm
and	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
separable	O
Fréchet	O
–	O
Montel	O
space	O
are	O
Souslin	O
spaces	O
.	O
</s>
<s>
The	O
Borel	B-Algorithm
graph	I-Algorithm
theorem	I-Algorithm
states	O
:	O
</s>
<s>
Let	O
and	O
be	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
and	O
let	O
be	O
linear	O
.	O
</s>
<s>
Also	O
,	O
every	O
Polish	O
,	O
Souslin	O
,	O
and	O
reflexive	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
K-analytic	O
as	O
is	O
the	O
weak	O
dual	O
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Let	O
and	O
be	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
spaces	O
and	O
let	O
be	O
linear	O
.	O
</s>
