<s>
In	O
mathematics	O
,	O
particularly	O
in	O
functional	B-Application
analysis	I-Application
,	O
a	O
bornological	B-Algorithm
space	I-Algorithm
is	O
a	O
type	O
of	O
space	O
which	O
,	O
in	O
some	O
sense	O
,	O
possesses	O
the	O
minimum	O
amount	O
of	O
structure	O
needed	O
to	O
address	O
questions	O
of	O
boundedness	B-Algorithm
of	O
sets	B-Algorithm
and	O
linear	O
maps	O
,	O
in	O
the	O
same	O
way	O
that	O
a	O
topological	O
space	O
possesses	O
the	O
minimum	O
amount	O
of	O
structure	O
needed	O
to	O
address	O
questions	O
of	O
continuity	O
.	O
</s>
<s>
Bornological	B-Algorithm
spaces	I-Algorithm
are	O
distinguished	O
by	O
the	O
property	O
that	O
a	O
linear	O
map	O
from	O
a	O
bornological	B-Algorithm
space	I-Algorithm
into	O
any	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
a	O
bounded	B-Algorithm
linear	O
operator	O
.	O
</s>
<s>
Bornological	B-Algorithm
spaces	I-Algorithm
were	O
first	O
studied	O
by	O
George	O
Mackey	O
.	O
</s>
<s>
The	O
name	O
was	O
coined	O
by	O
Bourbaki	O
after	O
,	O
the	O
French	O
word	O
for	O
"	O
bounded	B-Algorithm
"	O
.	O
</s>
<s>
A	O
subset	O
of	O
is	O
bounded	B-Algorithm
in	O
the	O
product	O
bornology	O
if	O
and	O
only	O
if	O
its	O
image	O
under	O
the	O
canonical	O
projections	O
onto	O
and	O
are	O
both	O
bounded	B-Algorithm
.	O
</s>
<s>
If	O
in	O
addition	O
is	O
a	O
bijection	O
and	O
is	O
also	O
bounded	B-Algorithm
then	O
is	O
called	O
a	O
.	O
</s>
<s>
if	O
the	O
sum	O
of	O
two	O
bounded	B-Algorithm
sets	I-Algorithm
is	O
bounded	B-Algorithm
,	O
etc	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
and	O
is	O
a	O
bornology	O
on	O
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
Finite	O
sums	O
and	O
balanced	O
hulls	O
of	O
-bounded	O
sets	O
are	O
-bounded	O
;	O
</s>
<s>
The	O
scalar	O
multiplication	O
map	O
defined	O
by	O
and	O
the	O
addition	O
map	O
defined	O
by	O
are	O
both	O
bounded	B-Algorithm
when	O
their	O
domains	O
carry	O
their	O
product	O
bornologies	O
(	O
i.e.	O
</s>
<s>
they	O
map	O
bounded	B-Algorithm
subsets	I-Algorithm
to	O
bounded	B-Algorithm
subsets	I-Algorithm
)	O
.	O
</s>
<s>
Usually	O
,	O
is	O
either	O
the	O
real	O
or	O
complex	O
numbers	O
,	O
in	O
which	O
case	O
a	O
vector	O
bornology	O
on	O
will	O
be	O
called	O
a	O
if	O
has	O
a	O
base	O
consisting	O
of	O
convex	O
sets	B-Algorithm
.	O
</s>
<s>
A	O
subset	O
of	O
is	O
called	O
and	O
a	O
if	O
it	O
absorbs	B-Algorithm
every	O
bounded	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
In	O
a	O
vector	O
bornology	O
,	O
is	O
bornivorous	B-Algorithm
if	O
it	O
absorbs	B-Algorithm
every	O
bounded	B-Algorithm
balanced	O
set	O
and	O
in	O
a	O
convex	O
vector	O
bornology	O
is	O
bornivorous	B-Algorithm
if	O
it	O
absorbs	B-Algorithm
every	O
bounded	B-Algorithm
disk	O
.	O
</s>
<s>
Two	O
TVS	O
topologies	O
on	O
the	O
same	O
vector	O
space	O
have	O
that	O
same	O
bounded	B-Algorithm
subsets	I-Algorithm
if	O
and	O
only	O
if	O
they	O
have	O
the	O
same	O
bornivores	B-Algorithm
.	O
</s>
<s>
Every	O
bornivorous	B-Algorithm
subset	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
,	O
then	O
an	O
absorbing	B-Algorithm
disk	O
in	O
is	O
bornivorous	B-Algorithm
(	O
resp	O
.	O
</s>
<s>
infrabornivorous	B-Algorithm
)	O
if	O
and	O
only	O
if	O
its	O
Minkowski	O
functional	O
is	O
locally	O
bounded	B-Algorithm
(	O
resp	O
.	O
</s>
<s>
If	O
is	O
a	O
convex	O
vector	O
bornology	O
on	O
a	O
vector	O
space	O
then	O
the	O
collection	O
of	O
all	O
convex	O
balanced	O
subsets	O
of	O
that	O
are	O
bornivorous	B-Algorithm
forms	O
a	O
neighborhood	O
basis	O
at	O
the	O
origin	O
for	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
called	O
the	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
with	O
a	O
continuous	B-Algorithm
dual	O
is	O
called	O
a	O
if	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
holds	O
:	O
</s>
<s>
Every	O
bounded	B-Algorithm
linear	O
operator	O
from	O
into	O
another	O
TVS	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
Every	O
bounded	B-Algorithm
linear	O
operator	O
from	O
into	O
a	O
complete	B-Algorithm
metrizable	I-Algorithm
TVS	I-Algorithm
is	O
continuous	B-Algorithm
.	O
</s>
<s>
Every	O
knot	O
in	O
a	O
bornivorous	B-Algorithm
string	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Every	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
is	O
quasi-bornological	O
.	O
</s>
<s>
A	O
TVS	O
in	O
which	O
every	O
bornivorous	B-Algorithm
set	I-Algorithm
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
is	O
a	O
quasi-bornological	O
space	O
.	O
</s>
<s>
If	O
is	O
a	O
quasi-bornological	O
TVS	O
then	O
the	O
finest	B-Algorithm
locally	I-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
that	O
is	O
coarser	O
than	O
makes	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
if	O
its	O
topology	O
can	O
be	O
recovered	O
from	O
its	O
bornology	O
in	O
a	O
natural	O
way	O
.	O
</s>
<s>
Every	O
locally	B-Algorithm
convex	I-Algorithm
quasi-bornological	O
space	O
is	O
bornological	B-Algorithm
but	O
there	O
exist	O
bornological	B-Algorithm
spaces	I-Algorithm
that	O
are	O
quasi-bornological	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
with	O
a	O
continuous	B-Algorithm
dual	O
is	O
called	O
a	O
if	O
it	O
is	O
locally	B-Algorithm
convex	I-Algorithm
and	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
holds	O
:	O
</s>
<s>
Every	O
convex	O
,	O
balanced	O
,	O
and	O
bornivorous	B-Algorithm
set	I-Algorithm
in	O
is	O
a	O
neighborhood	O
of	O
zero	O
.	O
</s>
<s>
Every	O
bounded	B-Algorithm
linear	O
operator	O
from	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
Recall	O
that	O
a	O
linear	O
map	O
is	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
it	O
maps	O
any	O
sequence	O
converging	O
to	O
in	O
the	O
domain	O
to	O
a	O
bounded	B-Algorithm
subset	O
of	O
the	O
codomain	O
.	O
</s>
<s>
In	O
particular	O
,	O
any	O
linear	O
map	O
that	O
is	O
sequentially	O
continuous	B-Algorithm
at	O
the	O
origin	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
Every	O
bounded	B-Algorithm
linear	O
operator	O
from	O
into	O
a	O
seminormed	O
space	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
Every	O
bounded	B-Algorithm
linear	O
operator	O
from	O
into	O
a	O
Banach	O
space	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
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>
The	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
induced	O
by	O
the	O
von	O
Neumann	O
bornology	O
on	O
is	O
the	O
same	O
as	O
'	O
s	O
given	O
topology	O
.	O
</s>
<s>
Every	O
bounded	B-Algorithm
seminorm	O
on	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
is	O
the	O
inductive	O
limit	O
of	O
the	O
normed	O
spaces	O
as	O
varies	O
over	O
the	O
closed	O
and	O
bounded	B-Algorithm
disks	O
of	O
(	O
or	O
as	O
varies	O
over	O
the	O
bounded	B-Algorithm
disks	O
of	O
)	O
.	O
</s>
<s>
carries	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
and	O
all	O
bounded	B-Algorithm
linear	O
functionals	O
on	O
are	O
continuous	B-Algorithm
.	O
</s>
<s>
is	O
or	O
,	O
which	O
means	O
that	O
every	O
convex	O
and	O
bornivorous	B-Algorithm
subset	O
of	O
is	O
sequentially	O
open	O
.	O
</s>
<s>
Every	O
sequentially	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
from	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
continuous	B-Algorithm
,	O
where	O
recall	O
that	O
a	O
linear	O
operator	O
is	O
sequentially	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
sequentially	O
continuous	B-Algorithm
at	O
the	O
origin	O
.	O
</s>
<s>
Thus	O
for	O
linear	O
maps	O
from	O
a	O
bornological	B-Algorithm
space	I-Algorithm
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
continuity	O
is	O
equivalent	O
to	O
sequential	O
continuity	O
at	O
the	O
origin	O
.	O
</s>
<s>
Any	O
linear	O
map	O
from	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
that	O
maps	O
null	O
sequences	O
in	O
to	O
bounded	B-Algorithm
subsets	I-Algorithm
of	O
is	O
necessarily	O
continuous	B-Algorithm
.	O
</s>
<s>
is	O
bornological.	O
"	O
</s>
<s>
The	O
following	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
are	O
all	O
bornological	B-Algorithm
:	O
</s>
<s>
Any	O
locally	B-Algorithm
convex	I-Algorithm
pseudometrizable	B-Algorithm
TVS	I-Algorithm
is	O
bornological	B-Algorithm
.	O
</s>
<s>
Thus	O
every	O
normed	O
space	O
and	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
bornological	B-Algorithm
.	O
</s>
<s>
Any	O
strict	O
inductive	O
limit	O
of	O
bornological	B-Algorithm
spaces	I-Algorithm
,	O
in	O
particular	O
any	O
strict	B-Algorithm
LF-space	I-Algorithm
,	O
is	O
bornological	B-Algorithm
.	O
</s>
<s>
This	O
shows	O
that	O
there	O
are	O
bornological	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
metrizable	O
.	O
</s>
<s>
A	O
countable	O
product	O
of	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
spaces	I-Algorithm
is	O
bornological	B-Algorithm
.	O
</s>
<s>
Quotients	O
of	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
spaces	I-Algorithm
are	O
bornological	B-Algorithm
.	O
</s>
<s>
The	O
direct	O
sum	O
and	O
inductive	O
limit	O
of	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
spaces	I-Algorithm
is	O
bornological	B-Algorithm
.	O
</s>
<s>
Fréchet	B-Algorithm
Montel	B-Algorithm
spaces	I-Algorithm
have	O
bornological	B-Algorithm
strong	B-Algorithm
duals	I-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
every	O
reflexive	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
bornological	B-Algorithm
.	O
</s>
<s>
If	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
separable	O
,	O
then	O
it	O
is	O
bornological	B-Algorithm
.	O
</s>
<s>
A	O
vector	O
subspace	O
of	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
that	O
has	O
finite	O
codimension	O
in	O
is	O
bornological	B-Algorithm
.	O
</s>
<s>
The	O
finest	B-Algorithm
locally	I-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
a	O
vector	O
space	O
is	O
bornological	B-Algorithm
.	O
</s>
<s>
There	O
exists	O
a	O
bornological	B-Algorithm
LB-space	B-Algorithm
whose	O
strong	O
bidual	O
is	O
bornological	B-Algorithm
.	O
</s>
<s>
A	O
closed	O
vector	O
subspace	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
is	O
not	O
necessarily	O
bornological	B-Algorithm
.	O
</s>
<s>
There	O
exists	O
a	O
closed	O
vector	O
subspace	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
that	O
is	O
complete	B-Algorithm
(	O
and	O
so	O
sequentially	O
complete	B-Algorithm
)	O
but	O
neither	O
barrelled	B-Algorithm
nor	O
bornological	B-Algorithm
.	O
</s>
<s>
Bornological	B-Algorithm
spaces	I-Algorithm
need	O
not	O
be	O
barrelled	B-Algorithm
and	O
barrelled	B-Algorithm
spaces	I-Algorithm
need	O
not	O
be	O
bornological	B-Algorithm
.	O
</s>
<s>
Because	O
every	O
locally	B-Algorithm
convex	I-Algorithm
ultrabornological	B-Algorithm
space	I-Algorithm
is	O
barrelled	B-Algorithm
,	O
it	O
follows	O
that	O
a	O
bornological	B-Algorithm
space	I-Algorithm
is	O
not	O
necessarily	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
is	O
complete	B-Algorithm
.	O
</s>
<s>
Every	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
is	O
infrabarrelled	B-Algorithm
.	O
</s>
<s>
Every	O
Hausdorff	O
sequentially	O
complete	B-Algorithm
bornological	B-Algorithm
TVS	O
is	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
Thus	O
every	O
compete	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
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>
Every	O
Hausdorff	O
bornological	B-Algorithm
space	I-Algorithm
is	O
quasi-barrelled	B-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
bornological	B-Algorithm
spaces	I-Algorithm
are	O
Mackey	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Every	O
quasi-complete	B-Algorithm
(	O
i.e.	O
</s>
<s>
all	O
closed	O
and	O
bounded	B-Algorithm
subsets	I-Algorithm
are	O
complete	B-Algorithm
)	O
bornological	B-Algorithm
space	I-Algorithm
is	O
barrelled	B-Algorithm
.	O
</s>
<s>
There	O
exist	O
,	O
however	O
,	O
bornological	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
barrelled	B-Algorithm
.	O
</s>
<s>
Every	O
bornological	B-Algorithm
space	I-Algorithm
is	O
the	O
inductive	O
limit	O
of	O
normed	O
spaces	O
(	O
and	O
Banach	O
spaces	O
if	O
the	O
space	O
is	O
also	O
quasi-complete	B-Algorithm
)	O
.	O
</s>
<s>
Let	O
be	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
with	O
continuous	B-Algorithm
dual	O
Then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
is	O
bornological	B-Algorithm
.	O
</s>
<s>
is	O
quasi-barrelled	B-Algorithm
.	O
</s>
<s>
is	O
barrelled	B-Algorithm
.	O
</s>
<s>
is	O
a	O
distinguished	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
linear	O
map	O
between	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
and	O
if	O
is	O
bornological	B-Algorithm
,	O
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
is	O
continuous	B-Algorithm
.	O
</s>
<s>
is	O
sequentially	O
continuous	B-Algorithm
.	O
</s>
<s>
For	O
every	O
set	O
that	O
's	O
bounded	B-Algorithm
in	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
and	O
that	O
the	O
space	O
of	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
is	O
endowed	O
with	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	I-Algorithm
bounded	I-Algorithm
subsets	I-Algorithm
of	O
If	O
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
and	O
if	O
is	O
complete	B-Algorithm
then	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
is	O
complete	B-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
need	O
not	O
be	O
bornological	B-Algorithm
.	O
</s>
<s>
In	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
,	O
every	O
convex	O
bornivorous	B-Algorithm
set	I-Algorithm
is	O
a	O
neighborhood	O
of	O
(	O
is	O
required	O
to	O
be	O
a	O
disk	O
)	O
.	O
</s>
<s>
Every	O
bornivorous	B-Algorithm
subset	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Closed	O
vector	O
subspaces	O
of	O
bornological	B-Algorithm
space	I-Algorithm
need	O
not	O
be	O
bornological	B-Algorithm
.	O
</s>
<s>
A	O
disk	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
called	O
if	O
it	O
absorbs	B-Algorithm
all	O
Banach	B-Algorithm
disks	I-Algorithm
.	O
</s>
<s>
If	O
is	O
locally	B-Algorithm
convex	I-Algorithm
and	O
Hausdorff	O
,	O
then	O
a	O
disk	O
is	O
infrabornivorous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
absorbs	B-Algorithm
all	O
compact	O
disks	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
called	O
if	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
hold	O
:	O
</s>
<s>
Every	O
infrabornivorous	B-Algorithm
disk	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
seminorm	O
on	O
that	O
is	O
bounded	B-Algorithm
on	O
each	O
Banach	B-Algorithm
disk	I-Algorithm
is	O
necessarily	O
continuous	B-Algorithm
.	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	B-Algorithm
on	O
each	O
Banach	B-Algorithm
disk	I-Algorithm
then	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
For	O
every	O
Banach	O
space	O
and	O
every	O
linear	O
map	O
if	O
is	O
bounded	B-Algorithm
on	O
each	O
Banach	B-Algorithm
disk	I-Algorithm
then	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
The	O
finite	O
product	O
of	O
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>
