<s>
In	O
mathematics	O
,	O
especially	O
functional	B-Application
analysis	I-Application
,	O
a	O
bornology	B-Algorithm
on	O
a	O
vector	O
space	O
over	O
a	O
field	O
where	O
has	O
a	O
bornology	B-Algorithm
ℬ	O
,	O
is	O
called	O
a	O
vector	B-Algorithm
bornology	I-Algorithm
if	O
makes	O
the	O
vector	O
space	O
operations	O
into	O
bounded	B-Algorithm
maps	O
.	O
</s>
<s>
A	O
subset	O
of	O
is	O
bounded	B-Algorithm
in	O
the	O
product	O
bornology	B-Algorithm
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>
A	O
bornology	B-Algorithm
on	O
is	O
called	O
a	O
if	O
it	O
is	O
stable	O
under	O
vector	O
addition	O
,	O
scalar	O
multiplication	O
,	O
and	O
the	O
formation	O
of	O
balanced	O
hulls	O
(	O
i.e.	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
and	O
is	O
a	O
bornology	B-Algorithm
on	O
then	O
the	O
following	O
are	O
equivalent	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	B-Algorithm
(	O
i.e.	O
</s>
<s>
A	O
vector	B-Algorithm
bornology	I-Algorithm
is	O
called	O
a	O
if	O
it	O
is	O
stable	O
under	O
the	O
formation	O
of	O
convex	O
hulls	O
(	O
i.e.	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	B-Algorithm
bornology	I-Algorithm
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	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
then	O
the	O
set	O
of	O
all	O
bounded	B-Algorithm
subsets	O
of	O
from	O
a	O
vector	B-Algorithm
bornology	I-Algorithm
on	O
called	O
the	O
,	O
the	O
,	O
or	O
simply	O
the	O
of	O
and	O
is	O
referred	O
to	O
as	O
.	O
</s>
<s>
Unless	O
indicated	O
otherwise	O
,	O
it	O
is	O
always	O
assumed	O
that	O
the	O
real	O
or	O
complex	O
numbers	O
are	O
endowed	O
with	O
the	O
usual	O
bornology	B-Algorithm
.	O
</s>
<s>
Let	O
denote	O
all	O
those	O
subsets	O
of	O
that	O
are	O
convex	O
,	O
balanced	O
,	O
and	O
bornivorous	B-Algorithm
.	O
</s>
<s>
Then	O
forms	O
a	O
neighborhood	O
basis	O
at	O
the	O
origin	O
for	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
topology	O
.	O
</s>
<s>
The	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
topology	O
on	O
defined	O
by	O
the	O
family	O
of	O
seminorms	O
is	O
called	O
the	O
.	O
</s>
