<s>
In	O
mathematics	O
,	O
a	O
hypocontinuous	B-Algorithm
is	O
a	O
condition	O
on	O
bilinear	O
maps	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
that	O
is	O
weaker	O
than	O
continuity	O
but	O
stronger	O
than	O
separate	O
continuity	O
.	O
</s>
<s>
Many	O
important	O
bilinear	O
maps	O
that	O
are	O
not	O
continuous	O
are	O
,	O
in	O
fact	O
,	O
hypocontinuous	B-Algorithm
.	O
</s>
<s>
If	O
,	O
and	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
then	O
a	O
bilinear	O
map	O
is	O
called	O
hypocontinuous	B-Algorithm
if	O
the	O
following	O
two	O
conditions	O
hold	O
:	O
</s>
<s>
for	O
every	O
bounded	B-Algorithm
set	I-Algorithm
the	O
set	O
of	O
linear	O
maps	O
is	O
an	O
equicontinuous	O
subset	O
of	O
.	O
</s>
<s>
Theorem	O
:	O
Let	O
X	O
and	O
Y	O
be	O
barreled	B-Algorithm
spaces	I-Algorithm
and	O
let	O
Z	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Then	O
every	O
separately	O
continuous	O
bilinear	O
map	O
of	O
into	O
Z	O
is	O
hypocontinuous	B-Algorithm
.	O
</s>
<s>
If	O
X	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
barreled	B-Algorithm
space	I-Algorithm
over	O
the	O
field	O
,	O
then	O
the	O
bilinear	O
map	O
defined	O
by	O
is	O
hypocontinuous	B-Algorithm
.	O
</s>
