<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
quasibarrelled	B-Algorithm
spaces	I-Algorithm
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVS	O
)	O
for	O
which	O
every	O
bornivorous	B-Algorithm
barrelled	O
set	O
in	O
the	O
space	O
is	O
a	O
neighbourhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Quasibarrelled	B-Algorithm
spaces	I-Algorithm
are	O
studied	O
because	O
they	O
are	O
a	O
weakening	O
of	O
the	O
defining	O
condition	O
of	O
barrelled	B-Algorithm
spaces	I-Algorithm
,	O
for	O
which	O
a	O
form	O
of	O
the	O
Banach	O
–	O
Steinhaus	O
theorem	O
holds	O
.	O
</s>
<s>
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
bornivorous	B-Algorithm
if	O
it	O
absorbs	O
all	O
bounded	O
subsets	O
of	O
;	O
</s>
<s>
A	O
barrelled	O
set	O
or	O
a	O
barrel	O
in	O
a	O
TVS	O
is	O
a	O
set	O
which	O
is	O
convex	O
,	O
balanced	O
,	O
absorbing	B-Algorithm
and	O
closed	O
.	O
</s>
<s>
A	O
quasibarrelled	B-Algorithm
space	I-Algorithm
is	O
a	O
TVS	O
for	O
which	O
every	O
bornivorous	B-Algorithm
barrelled	O
set	O
in	O
the	O
space	O
is	O
a	O
neighbourhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
quasibarrelled	B-Algorithm
space	I-Algorithm
that	O
is	O
sequentially	B-Algorithm
complete	I-Algorithm
is	O
barrelled	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
quasibarrelled	B-Algorithm
space	I-Algorithm
is	O
a	O
Mackey	B-Algorithm
space	I-Algorithm
,	O
quasi-M-barrelled	O
,	O
and	O
countably	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
quasibarrelled	B-Algorithm
space	I-Algorithm
that	O
is	O
also	O
a	O
σ-barrelled	B-Algorithm
space	I-Algorithm
is	O
necessarily	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
reflexive	O
if	O
and	O
only	O
if	O
it	O
is	O
semireflexive	O
and	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
A	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
quasibarrelled	B-Algorithm
if	O
and	O
only	O
if	O
every	O
bounded	O
closed	O
linear	O
operator	O
from	O
into	O
a	O
complete	B-Algorithm
metrizable	I-Algorithm
TVS	I-Algorithm
is	O
continuous	O
.	O
</s>
<s>
For	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
with	O
continuous	O
dual	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
is	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
is	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
is	O
barrelled	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
is	O
bornological	B-Algorithm
.	O
</s>
<s>
Every	O
Hausdorff	O
barrelled	B-Algorithm
space	I-Algorithm
and	O
every	O
Hausdorff	O
bornological	B-Algorithm
space	I-Algorithm
is	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
Thus	O
,	O
every	O
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
Note	O
that	O
there	O
exist	O
quasibarrelled	B-Algorithm
spaces	I-Algorithm
that	O
are	O
neither	O
barrelled	O
nor	O
bornological	B-Algorithm
.	O
</s>
<s>
There	O
exist	O
Mackey	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
There	O
exist	O
distinguished	B-Algorithm
spaces	I-Algorithm
,	O
DF-spaces	B-Algorithm
,	O
and	O
-barrelled	O
spaces	O
that	O
are	O
not	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
distinguished	B-Algorithm
if	O
and	O
only	O
if	O
is	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
There	O
exists	O
a	O
DF-space	B-Algorithm
that	O
is	O
not	O
quasibarrelled	B-Algorithm
.	O
</s>
<s>
There	O
exists	O
a	O
quasibarrelled	B-Algorithm
DF-space	B-Algorithm
that	O
is	O
not	O
bornological	B-Algorithm
.	O
</s>
<s>
There	O
exists	O
a	O
quasibarrelled	B-Algorithm
space	I-Algorithm
that	O
is	O
not	O
a	O
σ-barrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
