<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
an	O
ultrabarrelled	B-Algorithm
space	I-Algorithm
is	O
a	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVS	O
)	O
for	O
which	O
every	O
ultrabarrel	B-Algorithm
is	O
a	O
neighbourhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
TVS	O
is	O
called	O
ultrabarrelled	B-Algorithm
if	O
every	O
ultrabarrel	B-Algorithm
in	O
is	O
a	O
neighbourhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
ultrabarrelled	B-Algorithm
space	I-Algorithm
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
ultrabarrelled	B-Algorithm
space	I-Algorithm
is	O
a	O
quasi-ultrabarrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Complete	O
and	O
metrizable	O
TVSs	O
are	O
ultrabarrelled	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
complete	O
locally	O
bounded	O
non-locally	O
convex	O
TVS	O
and	O
if	O
is	O
a	O
closed	O
balanced	O
and	O
bounded	O
neighborhood	O
of	O
the	O
origin	O
,	O
then	O
is	O
an	O
ultrabarrel	B-Algorithm
that	O
is	O
not	O
convex	O
and	O
has	O
a	O
defining	O
sequence	O
consisting	O
of	O
non-convex	O
sets	O
.	O
</s>
<s>
There	O
exist	O
barrelled	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
ultrabarrelled	B-Algorithm
.	O
</s>
<s>
There	O
exist	O
TVSs	O
that	O
are	O
complete	O
and	O
metrizable	O
(	O
and	O
thus	O
ultrabarrelled	B-Algorithm
)	O
but	O
not	O
barrelled	O
.	O
</s>
