<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	O
)	O
is	O
B-complete	O
or	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
if	O
every	O
subspace	O
is	O
closed	O
in	O
the	O
weak-*	O
topology	O
on	O
(	O
i.e.	O
</s>
<s>
B-completeness	O
is	O
related	O
to	O
-completeness	O
,	O
where	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
-complete	O
if	O
every	O
subspace	O
is	O
closed	O
in	O
whenever	O
is	O
closed	O
in	O
(	O
when	O
is	O
given	O
the	O
subspace	O
topology	O
from	O
)	O
for	O
each	O
equicontinuous	O
subset	O
.	O
</s>
<s>
Throughout	O
this	O
section	O
,	O
will	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	O
)	O
.	O
</s>
<s>
is	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
continuous	O
nearly	O
open	O
linear	O
map	O
of	O
into	O
any	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
a	O
topological	O
homomorphism	O
.	O
</s>
<s>
is	O
-complete	O
.	O
</s>
<s>
Every	O
continuous	O
biunivocal	O
,	O
nearly	O
open	O
linear	O
map	O
of	O
into	O
any	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
a	O
TVS-isomorphism	O
.	O
</s>
<s>
Every	O
Ptak	B-Algorithm
space	I-Algorithm
is	O
complete	B-Algorithm
.	O
</s>
<s>
However	O
,	O
there	O
exist	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
that	O
are	O
not	O
Ptak	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
nearly	O
open	O
linear	O
map	O
whose	O
domain	O
is	O
dense	O
in	O
a	O
-complete	O
space	O
and	O
whose	O
range	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
is	O
injective	O
or	O
if	O
is	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
then	O
is	O
an	O
open	O
map	O
.	O
</s>
<s>
There	O
exist	O
Br-complete	O
spaces	O
that	O
are	O
not	O
B-complete	O
.	O
</s>
<s>
Every	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
strong	O
dual	O
of	O
a	O
reflexive	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
closed	O
vector	O
subspace	O
of	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
(	O
resp	O
.	O
</s>
<s>
a	O
Br-complete	O
space	O
)	O
is	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
(	O
resp	O
.	O
</s>
<s>
a	O
-complete	O
space	O
)	O
.	O
</s>
<s>
and	O
every	O
Hausdorff	O
quotient	O
of	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
is	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
every	O
Hausdorff	O
quotient	O
of	O
a	O
TVS	O
is	O
a	O
Br-complete	O
space	O
then	O
is	O
a	O
B-complete	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
such	O
that	O
there	O
exists	O
a	O
continuous	O
nearly	O
open	O
surjection	O
from	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
,	O
then	O
is	O
a	O
Ptak	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
a	O
TVS	O
has	O
a	O
closed	O
hyperplane	O
that	O
is	O
B-complete	O
(	O
resp	O
.	O
</s>
<s>
Br-complete	O
)	O
then	O
is	O
B-complete	O
(	O
resp	O
.	O
</s>
<s>
Br-complete	O
)	O
.	O
</s>
