<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
(	O
also	O
written	O
barreled	B-Algorithm
space	I-Algorithm
)	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
for	O
which	O
every	O
barrelled	B-Algorithm
set	I-Algorithm
in	O
the	O
space	O
is	O
a	O
neighbourhood	O
for	O
the	O
zero	O
vector	O
.	O
</s>
<s>
A	O
barrelled	B-Algorithm
set	I-Algorithm
or	O
a	O
barrel	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
a	O
set	O
that	O
is	O
convex	O
,	O
balanced	O
,	O
absorbing	B-Algorithm
,	O
and	O
closed	O
.	O
</s>
<s>
Barrelled	B-Algorithm
spaces	I-Algorithm
are	O
studied	O
because	O
a	O
form	O
of	O
the	O
Banach	O
–	O
Steinhaus	O
theorem	O
still	O
holds	O
for	O
them	O
.	O
</s>
<s>
Barrelled	B-Algorithm
spaces	I-Algorithm
were	O
introduced	O
by	O
.	O
</s>
<s>
A	O
or	O
a	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
a	O
subset	O
that	O
is	O
a	O
closed	O
absorbing	B-Algorithm
disk	O
;	O
that	O
is	O
,	O
a	O
barrel	O
is	O
a	O
convex	O
,	O
balanced	O
,	O
closed	O
,	O
and	O
absorbing	B-Algorithm
subset	O
.	O
</s>
<s>
If	O
is	O
any	O
TVS	O
then	O
every	O
closed	O
convex	O
and	O
balanced	O
neighborhood	O
of	O
the	O
origin	O
is	O
necessarily	O
a	O
barrel	O
in	O
(	O
because	O
every	O
neighborhood	O
of	O
the	O
origin	O
is	O
necessarily	O
an	O
absorbing	B-Algorithm
subset	O
)	O
.	O
</s>
<s>
In	O
fact	O
,	O
every	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
has	O
a	O
neighborhood	O
basis	O
at	O
its	O
origin	O
consisting	O
entirely	O
of	O
barrels	O
.	O
</s>
<s>
However	O
,	O
in	O
general	O
,	O
there	O
exist	O
barrels	O
that	O
are	O
not	O
neighborhoods	O
of	O
the	O
origin	O
;	O
"	O
barrelled	B-Algorithm
spaces	I-Algorithm
"	O
are	O
exactly	O
those	O
TVSs	O
in	O
which	O
every	O
barrel	O
is	O
necessarily	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Every	O
finite	O
dimensional	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
so	O
examples	O
of	O
barrels	O
that	O
are	O
not	O
neighborhoods	O
of	O
the	O
origin	O
can	O
only	O
be	O
found	O
in	O
infinite	O
dimensional	O
spaces	O
.	O
</s>
<s>
The	O
closure	O
of	O
any	O
convex	O
,	O
balanced	O
,	O
and	O
absorbing	B-Algorithm
subset	O
is	O
a	O
barrel	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
closure	O
of	O
any	O
convex	O
(	O
respectively	O
,	O
any	O
balanced	O
,	O
any	O
absorbing	B-Algorithm
)	O
subset	O
has	O
this	O
same	O
property	O
.	O
</s>
<s>
Regardless	O
of	O
whether	O
is	O
a	O
real	O
or	O
complex	O
vector	O
space	O
,	O
every	O
barrel	O
in	O
is	O
necessarily	O
a	O
neighborhood	O
of	O
the	O
origin	O
(	O
so	O
is	O
an	O
example	O
of	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
Let	O
be	O
any	O
function	O
and	O
for	O
every	O
angle	O
let	O
denote	O
the	O
closed	O
line	O
segment	O
from	O
the	O
origin	O
to	O
the	O
point	O
Let	O
Then	O
is	O
always	O
an	O
absorbing	B-Algorithm
subset	O
of	O
(	O
a	O
real	O
vector	O
space	O
)	O
but	O
it	O
is	O
an	O
absorbing	B-Algorithm
subset	O
of	O
(	O
a	O
complex	O
vector	O
space	O
)	O
if	O
and	O
only	O
if	O
it	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
In	O
particular	O
,	O
barrels	O
in	O
are	O
exactly	O
those	O
closed	O
balls	O
centered	O
at	O
the	O
origin	O
with	O
radius	O
in	O
If	O
then	O
is	O
a	O
closed	O
subset	O
that	O
is	O
absorbing	B-Algorithm
in	O
but	O
not	O
absorbing	B-Algorithm
in	O
and	O
that	O
is	O
neither	O
convex	O
,	O
balanced	O
,	O
nor	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
By	O
an	O
appropriate	O
choice	O
of	O
the	O
function	O
it	O
is	O
also	O
possible	O
to	O
have	O
be	O
a	O
balanced	O
and	O
absorbing	B-Algorithm
subset	O
of	O
that	O
is	O
neither	O
closed	O
nor	O
convex	O
.	O
</s>
<s>
Let	O
be	O
a	O
pairing	B-Algorithm
and	O
let	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
consistent	O
with	O
duality	O
.	O
</s>
<s>
Suppose	O
is	O
a	O
vector	O
subspace	O
of	O
finite	O
codimension	O
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
If	O
is	O
a	O
barrel	O
(	O
resp	O
.	O
</s>
<s>
bornivorous	B-Algorithm
barrel	O
,	O
bornivorous	B-Algorithm
disk	O
)	O
in	O
then	O
there	O
exists	O
a	O
barrel	O
(	O
resp	O
.	O
</s>
<s>
If	O
is	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
with	O
continuous	B-Algorithm
dual	O
space	O
then	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
This	O
definition	O
is	O
similar	O
to	O
a	O
characterization	O
of	O
Baire	O
TVSs	O
proved	O
by	O
Saxon	O
[1974],	O
who	O
proved	O
that	O
a	O
TVS	O
with	O
a	O
topology	O
that	O
is	O
not	O
the	O
indiscrete	O
topology	O
is	O
a	O
Baire	O
space	O
if	O
and	O
only	O
if	O
every	O
absorbing	B-Algorithm
balanced	O
subset	O
is	O
a	O
neighborhood	O
of	O
point	O
of	O
(	O
not	O
necessarily	O
the	O
origin	O
)	O
.	O
</s>
<s>
For	O
any	O
F-space	B-Algorithm
every	O
pointwise	O
bounded	O
subset	O
of	O
is	O
equicontinuous	O
.	O
</s>
<s>
An	O
F-space	B-Algorithm
is	O
a	O
complete	B-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
Every	O
closed	O
linear	O
operator	O
from	O
into	O
a	O
complete	B-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
continuous	B-Algorithm
.	O
</s>
<s>
If	O
is	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
this	O
list	O
may	O
be	O
extended	O
by	O
appending	O
:	O
</s>
<s>
For	O
any	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
every	O
pointwise	O
bounded	O
subset	O
of	O
is	O
equicontinuous	O
.	O
</s>
<s>
It	O
follows	O
from	O
the	O
above	O
two	O
characterizations	O
that	O
in	O
the	O
class	O
of	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
,	O
barrelled	B-Algorithm
spaces	I-Algorithm
are	O
exactly	O
those	O
for	O
which	O
the	O
uniform	O
boundedness	O
principal	O
holds	O
.	O
</s>
<s>
Every	O
-bounded	O
subset	O
of	O
the	O
continuous	B-Algorithm
dual	O
space	O
is	O
equicontinuous	O
(	O
this	O
provides	O
a	O
partial	O
converse	O
to	O
the	O
Banach-Steinhaus	O
theorem	O
)	O
..	O
</s>
<s>
Every	O
lower	O
semicontinuous	O
seminorm	O
on	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
Every	O
linear	O
map	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
almost	O
continuous	B-Algorithm
.	O
</s>
<s>
Every	O
surjective	O
linear	O
map	O
from	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
almost	B-Algorithm
open	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
this	O
list	O
may	O
be	O
extended	O
by	O
appending	O
:	O
</s>
<s>
Closed	B-Algorithm
graph	I-Algorithm
theorem	O
:	O
Every	O
closed	O
linear	O
operator	O
into	O
a	O
Banach	O
space	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
The	O
0-neighborhood	O
bases	O
in	O
and	O
the	O
fundamental	O
families	O
of	O
bounded	O
sets	O
in	O
correspond	O
to	O
each	O
other	O
by	O
polarity	B-Algorithm
.	O
</s>
<s>
If	O
is	O
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
then	O
this	O
list	O
may	O
be	O
extended	O
by	O
appending	O
:	O
</s>
<s>
For	O
any	O
complete	B-Algorithm
metrizable	B-Algorithm
TVS	I-Algorithm
every	O
pointwise	O
bounded	O
in	O
is	O
equicontinuous	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
then	O
this	O
list	O
may	O
be	O
extended	O
by	O
appending	O
:	O
</s>
<s>
(	O
)	O
:	O
The	O
weak*	O
topology	O
on	O
is	O
sequentially	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
Each	O
of	O
the	O
following	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
is	O
barreled	O
:	O
</s>
<s>
Consequently	O
,	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
that	O
is	O
of	O
the	O
second	O
category	O
in	O
itself	O
is	O
barrelled	O
.	O
</s>
<s>
F-spaces	B-Algorithm
,	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
Banach	O
spaces	O
,	O
and	O
Hilbert	O
spaces	O
.	O
</s>
<s>
Complete	B-Algorithm
pseudometrizable	B-Algorithm
TVSs	O
.	O
</s>
<s>
Montel	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Strong	B-Algorithm
dual	I-Algorithm
spaces	I-Algorithm
of	O
Montel	B-Algorithm
spaces	I-Algorithm
(	O
since	O
they	O
are	O
necessarily	O
Montel	B-Algorithm
spaces	I-Algorithm
)	O
.	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	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
sequentially	B-Algorithm
complete	I-Algorithm
quasibarrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
quasi-complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
infrabarrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
TVS	O
is	O
called	O
quasi-complete	B-Algorithm
if	O
every	O
closed	O
and	O
bounded	O
subset	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
Thus	O
the	O
completion	O
of	O
a	O
barreled	B-Algorithm
space	I-Algorithm
is	O
barrelled	O
.	O
</s>
<s>
A	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
with	O
a	O
dense	O
infrabarrelled	B-Algorithm
vector	O
subspace	O
.	O
</s>
<s>
Thus	O
the	O
completion	O
of	O
an	O
infrabarrelled	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
barrelled	O
.	O
</s>
<s>
A	O
vector	O
subspace	O
of	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
that	O
has	O
countable	O
codimensional	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
finite	O
codimensional	O
vector	O
subspace	O
of	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
is	O
barreled	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
ultrabarelled	O
TVS	O
.	O
</s>
<s>
A	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
such	O
that	O
every	O
weakly	O
bounded	O
subset	O
of	O
its	O
continuous	B-Algorithm
dual	O
space	O
is	O
equicontinuous	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
such	O
that	O
for	O
every	O
Banach	O
space	O
a	O
closed	O
linear	O
map	O
of	O
into	O
is	O
necessarily	O
continuous	B-Algorithm
.	O
</s>
<s>
A	O
product	O
of	O
a	O
family	O
of	O
barreled	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
direct	O
sum	O
and	O
the	O
inductive	O
limit	O
of	O
a	O
family	O
of	O
barrelled	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
A	O
quotient	O
of	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
Hausdorff	O
sequentially	B-Algorithm
complete	I-Algorithm
quasibarrelled	B-Algorithm
boundedly	O
summing	O
TVS	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
reflexive	O
space	O
is	O
barrelled	O
.	O
</s>
<s>
A	O
barrelled	B-Algorithm
space	I-Algorithm
need	O
not	O
be	O
Montel	B-Algorithm
,	O
complete	B-Algorithm
,	O
metrizable	B-Algorithm
,	O
unordered	O
Baire-like	O
,	O
nor	O
the	O
inductive	O
limit	O
of	O
Banach	O
spaces	O
.	O
</s>
<s>
However	O
,	O
they	O
are	O
all	O
infrabarrelled	B-Algorithm
.	O
</s>
<s>
A	O
closed	O
subspace	O
of	O
a	O
barreled	B-Algorithm
space	I-Algorithm
is	O
not	O
necessarily	O
countably	B-Algorithm
quasi-barreled	I-Algorithm
(	O
and	O
thus	O
not	O
necessarily	O
barrelled	O
)	O
.	O
</s>
<s>
There	O
exists	O
a	O
dense	O
vector	O
subspace	O
of	O
the	O
Fréchet	B-Algorithm
barrelled	B-Algorithm
space	I-Algorithm
that	O
is	O
not	O
barrelled	O
.	O
</s>
<s>
There	O
exist	O
complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
that	O
are	O
not	O
barrelled	O
.	O
</s>
<s>
The	O
finest	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
an	O
infinite-dimensional	O
vector	O
space	O
is	O
a	O
Hausdorff	O
barrelled	B-Algorithm
space	I-Algorithm
that	O
is	O
a	O
meagre	O
subset	O
of	O
itself	O
(	O
and	O
thus	O
not	O
a	O
Baire	O
space	O
)	O
.	O
</s>
<s>
The	O
importance	O
of	O
barrelled	B-Algorithm
spaces	I-Algorithm
is	O
due	O
mainly	O
to	O
the	O
following	O
results	O
.	O
</s>
<s>
Every	O
Hausdorff	O
barrelled	B-Algorithm
space	I-Algorithm
is	O
quasi-barrelled	B-Algorithm
.	O
</s>
<s>
A	O
linear	O
map	O
from	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
almost	O
continuous	B-Algorithm
.	O
</s>
<s>
A	O
linear	O
map	O
from	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
to	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
is	O
almost	B-Algorithm
open	I-Algorithm
.	O
</s>
<s>
A	O
separately	O
continuous	B-Algorithm
bilinear	O
map	O
from	O
a	O
product	O
of	O
barrelled	B-Algorithm
spaces	I-Algorithm
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
hypocontinuous	B-Algorithm
.	O
</s>
<s>
A	O
linear	O
map	O
with	O
a	O
closed	B-Algorithm
graph	I-Algorithm
from	O
a	O
barreled	O
TVS	O
into	O
a	O
-complete	O
TVS	O
is	O
necessarily	O
continuous	B-Algorithm
.	O
</s>
