<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
said	O
to	O
be	O
countably	O
barrelled	O
if	O
every	O
weakly	O
bounded	O
countable	O
union	O
of	O
equicontinuous	O
subsets	O
of	O
its	O
continuous	O
dual	O
space	O
is	O
again	O
equicontinuous	O
.	O
</s>
<s>
This	O
property	O
is	O
a	O
generalization	O
of	O
barrelled	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
A	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
countably	O
barrelled	O
if	O
and	O
only	O
if	O
each	O
barrel	B-Algorithm
in	O
X	O
that	O
is	O
equal	O
to	O
the	O
countable	O
intersection	O
of	O
closed	O
convex	O
balanced	O
neighborhoods	O
of	O
0	O
is	O
itself	O
a	O
neighborhood	O
of	O
0	O
.	O
</s>
<s>
Every	O
countably	B-Algorithm
barrelled	I-Algorithm
space	I-Algorithm
is	O
a	O
countably	B-Algorithm
quasibarrelled	I-Algorithm
space	I-Algorithm
,	O
a	O
σ-barrelled	B-Algorithm
space	I-Algorithm
,	O
a	O
σ-quasi-barrelled	B-Algorithm
space	I-Algorithm
,	O
and	O
a	O
sequentially	B-Algorithm
barrelled	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
An	O
H-space	O
is	O
a	O
TVS	O
whose	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
is	O
countably	O
barrelled	O
.	O
</s>
<s>
Every	O
countably	B-Algorithm
barrelled	I-Algorithm
space	I-Algorithm
is	O
a	O
σ-barrelled	B-Algorithm
space	I-Algorithm
and	O
every	O
σ-barrelled	B-Algorithm
space	I-Algorithm
is	O
sequentially	O
barrelled	O
.	O
</s>
<s>
Every	O
σ-barrelled	B-Algorithm
space	I-Algorithm
is	O
a	O
σ-quasi-barrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
quasi-barrelled	B-Algorithm
space	I-Algorithm
that	O
is	O
also	O
a	O
𝜎-barrelled	O
space	O
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
barrelled	B-Algorithm
space	I-Algorithm
is	O
countably	O
barrelled	O
.	O
</s>
<s>
However	O
,	O
there	O
exist	O
semi-reflexive	O
countably	B-Algorithm
barrelled	I-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
barrelled	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
distinguished	B-Algorithm
space	I-Algorithm
and	O
of	O
a	O
metrizable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
countably	O
barrelled	O
.	O
</s>
<s>
There	O
exist	O
σ-barrelled	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
countably	O
barrelled	O
.	O
</s>
<s>
There	O
exist	O
normed	O
DF-spaces	B-Algorithm
that	O
are	O
not	O
countably	O
barrelled	O
.	O
</s>
<s>
There	O
exists	O
a	O
quasi-barrelled	B-Algorithm
space	I-Algorithm
that	O
is	O
not	O
a	O
𝜎-barrelled	O
space	O
.	O
</s>
<s>
There	O
exist	O
σ-barrelled	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
Mackey	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
There	O
exist	O
σ-barrelled	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
countably	B-Algorithm
quasi-barrelled	I-Algorithm
spaces	I-Algorithm
and	O
thus	O
not	O
countably	O
barrelled	O
.	O
</s>
<s>
There	O
exist	O
sequentially	B-Algorithm
barrelled	I-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
σ-quasi-barrelled	O
.	O
</s>
<s>
There	O
exist	O
quasi-complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
that	O
are	O
not	O
sequentially	O
barrelled	O
.	O
</s>
