<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
subset	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
called	O
a	O
barrel	O
or	O
a	O
barrelled	B-Algorithm
set	I-Algorithm
if	O
it	O
is	O
closed	O
convex	O
balanced	O
and	O
absorbing	B-Algorithm
.	O
</s>
<s>
Barrelled	B-Algorithm
sets	I-Algorithm
play	O
an	O
important	O
role	O
in	O
the	O
definitions	O
of	O
several	O
classes	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
such	O
as	O
barrelled	B-Algorithm
spaces	I-Algorithm
.	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>
Let	O
be	O
a	O
subset	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
If	O
is	O
a	O
balanced	O
absorbing	B-Algorithm
subset	I-Algorithm
of	O
and	O
if	O
there	O
exists	O
a	O
sequence	O
of	O
balanced	O
absorbing	B-Algorithm
subsets	O
of	O
such	O
that	O
for	O
all	O
then	O
is	O
called	O
a	O
in	O
where	O
moreover	O
,	O
is	O
said	O
to	O
be	O
a(n )	O
:	O
</s>
<s>
Note	O
that	O
every	O
bornivorous	B-Algorithm
ultrabarrel	O
is	O
an	O
ultrabarrel	O
and	O
that	O
every	O
bornivorous	B-Algorithm
suprabarrel	O
is	O
a	O
suprabarrel	O
.	O
</s>
<s>
Every	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
has	O
a	O
neighbourhood	O
basis	O
consisting	O
of	O
barrelled	B-Algorithm
sets	I-Algorithm
,	O
although	O
the	O
space	O
itself	O
need	O
not	O
be	O
a	O
barreled	B-Algorithm
space	I-Algorithm
.	O
</s>
