<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
called	O
ultrabornological	B-Algorithm
if	O
every	O
bounded	O
linear	O
operator	O
from	O
into	O
another	O
TVS	O
is	O
necessarily	O
continuous	O
.	O
</s>
<s>
A	O
general	O
version	O
of	O
the	O
closed	O
graph	O
theorem	O
holds	O
for	O
ultrabornological	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Ultrabornological	B-Algorithm
spaces	I-Algorithm
were	O
introduced	O
by	O
Alexander	O
Grothendieck	O
(	O
Grothendieck	O
[	O
1955	O
,	O
p	O
.	O
17 ]	O
"	O
espace	O
du	O
type	O
( β	O
)	O
"	O
)	O
.	O
</s>
<s>
Let	O
be	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
.	O
</s>
<s>
A	O
linear	O
map	O
between	O
two	O
TVSs	O
is	O
called	O
infrabounded	O
if	O
it	O
maps	O
Banach	B-Algorithm
disks	I-Algorithm
to	O
bounded	O
disks	O
.	O
</s>
<s>
A	O
disk	O
in	O
a	O
TVS	O
is	O
called	O
infrabornivorous	B-Algorithm
if	O
it	O
satisfies	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
:	O
</s>
<s>
while	O
if	O
locally	B-Algorithm
convex	I-Algorithm
then	O
we	O
may	O
add	O
to	O
this	O
list	O
:	O
</s>
<s>
while	O
if	O
locally	B-Algorithm
convex	I-Algorithm
and	O
Hausdorff	O
then	O
we	O
may	O
add	O
to	O
this	O
list	O
:	O
</s>
<s>
absorbs	B-Algorithm
all	O
compact	O
disks	O
;	O
that	O
is	O
,	O
is	O
"	O
compactivorious	O
"	O
.	O
</s>
<s>
A	O
TVS	O
is	O
ultrabornological	B-Algorithm
if	O
it	O
satisfies	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
:	O
</s>
<s>
every	O
infrabornivorous	B-Algorithm
disk	O
in	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
;	O
</s>
<s>
while	O
if	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
we	O
may	O
add	O
to	O
this	O
list	O
:	O
</s>
<s>
every	O
bounded	O
linear	O
operator	O
from	O
into	O
a	O
complete	B-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
necessarily	O
continuous	O
;	O
</s>
<s>
every	O
infrabornivorous	B-Algorithm
disk	O
is	O
a	O
neighborhood	O
of	O
0	O
;	O
</s>
<s>
a	O
seminorm	O
on	O
that	O
is	O
bounded	O
on	O
each	O
Banach	B-Algorithm
disk	I-Algorithm
is	O
necessarily	O
continuous	O
;	O
</s>
<s>
for	O
every	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
every	O
linear	O
map	O
if	O
is	O
bounded	O
on	O
each	O
Banach	B-Algorithm
disk	I-Algorithm
then	O
is	O
continuous	O
;	O
</s>
<s>
for	O
every	O
Banach	O
space	O
and	O
every	O
linear	O
map	O
if	O
is	O
bounded	O
on	O
each	O
Banach	B-Algorithm
disk	I-Algorithm
then	O
is	O
continuous	O
.	O
</s>
<s>
while	O
if	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
we	O
may	O
add	O
to	O
this	O
list	O
:	O
</s>
<s>
Every	O
locally	B-Algorithm
convex	I-Algorithm
ultrabornological	B-Algorithm
space	I-Algorithm
is	O
barrelled	B-Algorithm
,	O
quasi-ultrabarrelled	B-Algorithm
space	I-Algorithm
,	O
and	O
a	O
bornological	B-Algorithm
space	I-Algorithm
but	O
there	O
exist	O
bornological	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
The	O
finite	O
product	O
of	O
locally	B-Algorithm
convex	I-Algorithm
ultrabornological	B-Algorithm
spaces	I-Algorithm
is	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
Inductive	O
limits	O
of	O
ultrabornological	B-Algorithm
spaces	I-Algorithm
are	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
Every	O
Hausdorff	O
sequentially	O
complete	B-Algorithm
bornological	B-Algorithm
space	I-Algorithm
is	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
Thus	O
every	O
complete	B-Algorithm
Hausdorff	O
bornological	B-Algorithm
space	I-Algorithm
is	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
every	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
complete	B-Algorithm
Schwartz	B-Algorithm
space	I-Algorithm
is	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
Every	O
Hausdorff	O
bornological	B-Algorithm
space	I-Algorithm
that	O
is	O
quasi-complete	B-Algorithm
is	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
There	O
exist	O
ultrabarrelled	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
There	O
exist	O
ultrabornological	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
ultrabarrelled	B-Algorithm
.	O
</s>
