<s>
In	O
mathematics	O
,	O
particularly	O
functional	B-Application
analysis	I-Application
,	O
spaces	O
of	O
linear	B-Architecture
maps	I-Architecture
between	O
two	O
vector	O
spaces	O
can	O
be	O
endowed	O
with	O
a	O
variety	O
of	O
topologies	O
.	O
</s>
<s>
Studying	O
space	B-Algorithm
of	I-Algorithm
linear	I-Algorithm
maps	I-Algorithm
and	O
these	O
topologies	O
can	O
give	O
insight	O
into	O
the	O
spaces	O
themselves	O
.	O
</s>
<s>
The	O
article	O
operator	B-Algorithm
topologies	I-Algorithm
discusses	O
topologies	B-Algorithm
on	I-Algorithm
spaces	I-Algorithm
of	I-Algorithm
linear	I-Algorithm
maps	I-Algorithm
between	O
normed	O
spaces	O
,	O
whereas	O
this	O
article	O
discusses	O
topologies	O
on	O
such	O
spaces	O
in	O
the	O
more	O
general	O
setting	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
.	O
</s>
<s>
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
not	O
necessarily	O
Hausdorff	O
or	O
locally	B-Algorithm
convex	I-Algorithm
)	O
.	O
</s>
<s>
is	O
a	O
vector	O
subspace	O
of	O
Because	O
is	O
just	O
a	O
set	O
that	O
is	O
not	O
yet	O
assumed	O
to	O
be	O
endowed	O
with	O
any	O
vector	O
space	O
structure	O
,	O
should	O
not	O
yet	O
be	O
assumed	O
to	O
consist	O
of	O
linear	B-Architecture
maps	I-Architecture
,	O
which	O
is	O
a	O
notation	O
that	O
currently	O
can	O
not	O
be	O
defined	O
.	O
</s>
<s>
The	O
following	O
sets	O
will	O
constitute	O
the	O
basic	O
open	O
subsets	O
of	O
topologies	B-Algorithm
on	I-Algorithm
spaces	I-Algorithm
of	I-Algorithm
linear	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
This	O
topology	O
does	O
not	O
depend	O
on	O
the	O
neighborhood	O
basis	O
that	O
was	O
chosen	O
and	O
it	O
is	O
known	O
as	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
the	O
sets	O
in	O
or	O
as	O
the	O
-topology	O
.	O
</s>
<s>
the	O
"	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
compact	O
sets	O
"	O
or	O
the	O
"	O
topology	O
of	O
compact	O
convergence	O
"	O
,	O
see	O
the	O
footnote	O
for	O
more	O
details	O
)	O
.	O
</s>
<s>
Then	O
is	O
an	O
absorbing	B-Algorithm
subset	O
of	O
if	O
and	O
only	O
if	O
for	O
all	O
absorbs	B-Algorithm
.	O
</s>
<s>
If	O
is	O
locally	B-Algorithm
convex	I-Algorithm
then	O
so	O
is	O
the	O
-topology	O
on	O
and	O
if	O
is	O
a	O
family	O
of	O
continuous	O
seminorms	O
generating	O
this	O
topology	O
on	O
then	O
the	O
-topology	O
is	O
induced	O
by	O
the	O
following	O
family	O
of	O
seminorms	O
:	O
</s>
<s>
If	O
is	O
Hausdorff	O
and	O
is	O
the	O
vector	O
subspace	O
of	O
consisting	O
of	O
all	O
continuous	O
maps	O
that	O
are	O
bounded	B-Algorithm
on	O
every	O
and	O
if	O
is	O
dense	O
in	O
then	O
the	O
-topology	O
on	O
is	O
Hausdorff	O
.	O
</s>
<s>
If	O
is	O
a	O
non-trivial	O
completely	O
regular	O
Hausdorff	O
topological	O
space	O
and	O
is	O
the	O
space	O
of	O
all	O
real	O
(	O
or	O
complex	O
)	O
valued	O
continuous	O
functions	O
on	O
the	O
topology	O
of	O
pointwise	O
convergence	O
on	O
is	O
metrizable	B-Algorithm
if	O
and	O
only	O
if	O
is	O
countable	O
.	O
</s>
<s>
Throughout	O
this	O
section	O
we	O
will	O
assume	O
that	O
and	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
will	O
denote	O
the	O
vector	O
space	O
of	O
all	O
continuous	O
linear	B-Architecture
maps	I-Architecture
from	O
into	O
If	O
is	O
given	O
the	O
-topology	O
inherited	O
from	O
then	O
this	O
space	O
with	O
this	O
topology	O
is	O
denoted	O
by	O
.	O
</s>
<s>
The	O
continuous	O
dual	O
space	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
over	O
the	O
field	O
(	O
which	O
we	O
will	O
assume	O
to	O
be	O
real	O
or	O
complex	O
numbers	O
)	O
is	O
the	O
vector	O
space	O
and	O
is	O
denoted	O
by	O
.	O
</s>
<s>
The	O
-topology	O
on	O
is	O
compatible	O
with	O
the	O
vector	O
space	O
structure	O
of	O
if	O
and	O
only	O
if	O
for	O
all	O
and	O
all	O
the	O
set	O
is	O
bounded	B-Algorithm
in	O
which	O
we	O
will	O
assume	O
to	O
be	O
the	O
case	O
for	O
the	O
rest	O
of	O
the	O
article	O
.	O
</s>
<s>
If	O
is	O
a	O
bornology	B-Algorithm
on	O
which	O
is	O
often	O
the	O
case	O
,	O
then	O
these	O
axioms	O
are	O
satisfied	O
.	O
</s>
<s>
If	O
is	O
a	O
saturated	B-Algorithm
family	I-Algorithm
of	O
bounded	B-Algorithm
subsets	O
of	O
then	O
these	O
axioms	O
are	O
also	O
satisfied	O
.	O
</s>
<s>
If	O
is	O
a	O
Mackey	O
space	O
then	O
is	O
complete	B-Algorithm
if	O
and	O
only	O
if	O
both	O
and	O
are	O
complete	B-Algorithm
.	O
</s>
<s>
If	O
is	O
barrelled	B-Algorithm
then	O
is	O
Hausdorff	O
and	O
quasi-complete	B-Algorithm
.	O
</s>
<s>
Let	O
and	O
be	O
TVSs	O
with	O
quasi-complete	B-Algorithm
and	O
assume	O
that	O
(	O
1	O
)	O
is	O
barreled	B-Algorithm
,	O
or	O
else	O
(	O
2	O
)	O
is	O
a	O
Baire	O
space	O
and	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
If	O
covers	O
then	O
every	O
closed	O
equicontinuous	O
subset	O
of	O
is	O
complete	B-Algorithm
in	O
and	O
is	O
quasi-complete	B-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
bornological	B-Algorithm
space	I-Algorithm
,	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
and	O
a	O
family	O
of	O
bounded	B-Algorithm
subsets	O
of	O
such	O
that	O
the	O
range	O
of	O
every	O
null	O
sequence	O
in	O
is	O
contained	O
in	O
some	O
If	O
is	O
quasi-complete	B-Algorithm
(	O
respectively	O
,	O
complete	B-Algorithm
)	O
then	O
so	O
is	O
.	O
</s>
<s>
is	O
bounded	B-Algorithm
in	O
;	O
</s>
<s>
For	O
every	O
is	O
bounded	B-Algorithm
in	O
;	O
</s>
<s>
If	O
is	O
a	O
collection	O
of	O
bounded	B-Algorithm
subsets	O
of	O
whose	O
union	O
is	O
total	O
in	O
then	O
every	O
equicontinuous	O
subset	O
of	O
is	O
bounded	B-Algorithm
in	O
the	O
-topology	O
.	O
</s>
<s>
Unfortunately	O
,	O
this	O
topology	O
is	O
also	O
sometimes	O
called	O
the	O
strong	O
operator	B-Algorithm
topology	I-Algorithm
,	O
which	O
may	O
lead	O
to	O
ambiguity	O
;	O
for	O
this	O
reason	O
,	O
this	O
article	O
will	O
avoid	O
referring	O
to	O
this	O
topology	O
by	O
this	O
name	O
.	O
</s>
<s>
A	O
subset	O
of	O
is	O
called	O
simply	O
bounded	B-Algorithm
or	O
weakly	O
bounded	B-Algorithm
if	O
it	O
is	O
bounded	B-Algorithm
in	O
.	O
</s>
<s>
So	O
in	O
particular	O
,	O
on	O
every	O
equicontinuous	O
subset	O
of	O
the	O
topology	O
of	O
pointwise	O
convergence	O
is	O
metrizable	B-Algorithm
.	O
</s>
<s>
Let	O
denote	O
the	O
space	O
of	O
all	O
functions	O
from	O
into	O
If	O
is	O
given	O
the	O
topology	O
of	O
pointwise	O
convergence	O
then	O
space	O
of	O
all	O
linear	B-Architecture
maps	I-Architecture
(	O
continuous	O
or	O
not	O
)	O
into	O
is	O
closed	O
in	O
.	O
</s>
<s>
Suppose	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
If	O
is	O
locally	B-Algorithm
convex	I-Algorithm
,	O
then	O
the	O
convex	O
balanced	O
hull	O
of	O
an	O
equicontinuous	O
subset	O
of	O
is	O
equicontinuous	O
.	O
</s>
<s>
Let	O
and	O
be	O
TVSs	O
and	O
assume	O
that	O
(	O
1	O
)	O
is	O
barreled	B-Algorithm
,	O
or	O
else	O
(	O
2	O
)	O
is	O
a	O
Baire	O
space	O
and	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
Then	O
every	O
simply	O
bounded	B-Algorithm
subset	O
of	O
is	O
equicontinuous	O
.	O
</s>
<s>
By	O
letting	O
be	O
the	O
set	O
of	O
all	O
compact	O
subsets	O
of	O
will	O
have	O
the	O
topology	O
of	O
compact	O
convergence	O
or	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
compact	O
sets	O
and	O
with	O
this	O
topology	O
is	O
denoted	O
by	O
.	O
</s>
<s>
If	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
or	O
a	O
LF-space	B-Algorithm
and	O
if	O
is	O
a	O
complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
space	O
then	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
and	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
,	O
then	O
and	O
have	O
identical	O
topologies	O
.	O
</s>
<s>
By	O
letting	O
be	O
the	O
set	O
of	O
all	O
bounded	B-Algorithm
subsets	O
of	O
will	O
have	O
the	O
topology	O
of	O
bounded	B-Algorithm
convergence	O
on	O
or	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
bounded	B-Algorithm
sets	I-Algorithm
and	O
with	O
this	O
topology	O
is	O
denoted	O
by	O
.	O
</s>
<s>
The	O
topology	O
of	O
bounded	B-Algorithm
convergence	O
on	O
has	O
the	O
following	O
properties	O
:	O
</s>
<s>
If	O
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
and	O
if	O
is	O
a	O
complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
space	O
then	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
is	O
a	O
normed	O
space	O
then	O
the	O
usual	O
norm	O
topology	O
on	O
the	O
continuous	O
dual	O
space	O
is	O
identical	O
to	O
the	O
topology	O
of	O
bounded	B-Algorithm
convergence	O
on	O
.	O
</s>
<s>
Every	O
equicontinuous	O
subset	O
of	O
is	O
bounded	B-Algorithm
in	O
.	O
</s>
<s>
If	O
is	O
a	O
TVS	O
whose	O
bounded	B-Algorithm
subsets	O
are	O
exactly	O
the	O
same	O
as	O
its	O
bounded	B-Algorithm
subsets	O
(	O
e.g.	O
</s>
<s>
if	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
)	O
,	O
then	O
a	O
-topology	O
on	O
(	O
as	O
defined	O
in	O
this	O
article	O
)	O
is	O
a	O
polar	B-Algorithm
topology	I-Algorithm
and	O
conversely	O
,	O
every	O
polar	B-Algorithm
topology	I-Algorithm
if	O
a	O
-topology	O
.	O
</s>
<s>
Consequently	O
,	O
in	O
this	O
case	O
the	O
results	O
mentioned	O
in	O
this	O
article	O
can	O
be	O
applied	O
to	O
polar	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
However	O
,	O
if	O
is	O
a	O
TVS	O
whose	O
bounded	B-Algorithm
subsets	O
are	O
exactly	O
the	O
same	O
as	O
its	O
bounded	B-Algorithm
subsets	O
,	O
then	O
the	O
notion	O
of	O
"	O
bounded	B-Algorithm
in	O
"	O
is	O
stronger	O
than	O
the	O
notion	O
of	O
"	O
-bounded	O
in	O
"	O
(	O
i.e.	O
</s>
<s>
bounded	B-Algorithm
in	O
implies	O
-bounded	O
in	O
)	O
so	O
that	O
a	O
-topology	O
on	O
(	O
as	O
defined	O
in	O
this	O
article	O
)	O
is	O
necessarily	O
a	O
polar	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
One	O
important	O
difference	O
is	O
that	O
polar	B-Algorithm
topologies	I-Algorithm
are	O
always	O
locally	B-Algorithm
convex	I-Algorithm
while	O
-topologies	O
need	O
not	O
be	O
.	O
</s>
<s>
Polar	B-Algorithm
topologies	I-Algorithm
have	O
stronger	O
results	O
than	O
the	O
more	O
general	O
topologies	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
described	O
in	O
this	O
article	O
and	O
we	O
refer	O
the	O
read	O
to	O
the	O
main	O
article	O
:	O
polar	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
We	O
list	O
here	O
some	O
of	O
the	O
most	O
common	O
polar	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
TVS	O
whose	O
bounded	B-Algorithm
subsets	O
are	O
the	O
same	O
as	O
its	O
weakly	O
bounded	B-Algorithm
subsets	O
.	O
</s>
<s>
Notation	O
:	O
If	O
denotes	O
a	O
polar	B-Algorithm
topology	I-Algorithm
on	O
then	O
endowed	O
with	O
this	O
topology	O
will	O
be	O
denoted	O
by	O
or	O
simply	O
(	O
e.g.	O
</s>
<s>
We	O
will	O
let	O
denote	O
the	O
space	O
of	O
separately	O
continuous	O
bilinear	O
maps	O
and	O
denote	O
the	O
space	O
of	O
continuous	O
bilinear	O
maps	O
,	O
where	O
and	O
are	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
over	O
the	O
same	O
field	O
(	O
either	O
the	O
real	O
or	O
complex	O
numbers	O
)	O
.	O
</s>
<s>
This	O
topology	O
is	O
known	O
as	O
the	O
-topology	O
or	O
as	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
the	O
products	O
of	O
.	O
</s>
<s>
If	O
both	O
and	O
consist	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
then	O
this	O
requirement	O
is	O
automatically	O
satisfied	O
if	O
we	O
are	O
topologizing	O
but	O
this	O
may	O
not	O
be	O
the	O
case	O
if	O
we	O
are	O
trying	O
to	O
topologize	O
.	O
</s>
<s>
The	O
-topology	O
on	O
will	O
be	O
compatible	O
with	O
the	O
vector	O
space	O
structure	O
of	O
if	O
both	O
and	O
consists	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
and	O
any	O
of	O
the	O
following	O
conditions	O
hold	O
:	O
</s>
<s>
and	O
are	O
barrelled	B-Algorithm
spaces	I-Algorithm
and	O
is	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
and	O
are	O
the	O
strong	B-Algorithm
duals	I-Algorithm
of	O
reflexive	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
is	O
normed	O
and	O
and	O
the	O
strong	B-Algorithm
duals	I-Algorithm
of	O
reflexive	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
and	O
let	O
and	O
be	O
the	O
collections	O
of	O
equicontinuous	O
subsets	O
of	O
and	O
,	O
respectively	O
.	O
</s>
<s>
Then	O
the	O
-topology	O
on	O
will	O
be	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
topology	O
.	O
</s>
<s>
If	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
spaces	O
then	O
is	O
complete	B-Algorithm
if	O
and	O
only	O
if	O
both	O
and	O
are	O
complete	B-Algorithm
.	O
</s>
