<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
a	O
polar	B-Algorithm
topology	I-Algorithm
,	O
topology	O
of	O
-convergence	O
or	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
the	O
sets	O
of	O
is	O
a	O
method	O
to	O
define	O
locally	B-Algorithm
convex	I-Algorithm
topologies	I-Algorithm
on	O
the	O
vector	O
spaces	O
of	O
a	O
pairing	B-Algorithm
.	O
</s>
<s>
A	O
dual	B-Algorithm
pair	I-Algorithm
or	O
dual	B-Algorithm
system	I-Algorithm
is	O
a	O
pairing	B-Algorithm
satisfying	O
the	O
following	O
two	O
separation	O
axioms	O
:	O
</s>
<s>
The	O
polar	B-Algorithm
is	O
a	O
convex	O
balanced	O
set	O
containing	O
the	O
origin	O
.	O
</s>
<s>
Convention	O
and	O
Definition	O
:	O
Given	O
any	O
definition	O
for	O
a	O
pairing	B-Algorithm
one	O
obtains	O
a	O
dual	O
definition	O
by	O
applying	O
it	O
to	O
the	O
pairing	B-Algorithm
If	O
the	O
definition	O
depends	O
on	O
the	O
order	O
of	O
and	O
(	O
e.g.	O
</s>
<s>
Convention	O
:	O
Adhering	O
to	O
common	O
practice	O
,	O
unless	O
clarity	O
is	O
needed	O
,	O
whenever	O
a	O
definition	O
(	O
or	O
result	O
)	O
is	O
given	O
for	O
a	O
pairing	B-Algorithm
then	O
mention	O
the	O
corresponding	O
dual	O
definition	O
(	O
or	O
result	O
)	O
will	O
be	O
omitted	O
but	O
it	O
may	O
nevertheless	O
be	O
used	O
.	O
</s>
<s>
where	O
and	O
is	O
in	O
fact	O
the	O
Minkowski	O
functional	O
of	O
Because	O
of	O
this	O
,	O
the	O
-topology	O
on	O
is	O
always	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
The	O
results	O
in	O
the	O
article	O
Topologies	B-Algorithm
on	I-Algorithm
spaces	I-Algorithm
of	I-Algorithm
linear	I-Algorithm
maps	I-Algorithm
can	O
be	O
applied	O
to	O
polar	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
the	O
topology	O
generated	O
by	O
-complete	O
and	O
bounded	B-Algorithm
disks	O
)	O
or	O
if	O
is	O
not	O
Hausdorff	O
.	O
</s>
<s>
If	O
more	O
than	O
one	O
collection	O
of	O
subsets	O
appears	O
the	O
same	O
row	O
in	O
the	O
left-most	O
column	O
then	O
that	O
means	O
that	O
the	O
same	O
polar	B-Algorithm
topology	I-Algorithm
is	O
generated	O
by	O
these	O
collections	O
.	O
</s>
<s>
If	O
and	O
are	O
vector	O
spaces	O
over	O
the	O
complex	O
numbers	O
(	O
which	O
implies	O
that	O
is	O
complex	O
valued	O
)	O
then	O
let	O
and	O
denote	O
these	O
spaces	O
when	O
they	O
are	O
considered	O
as	O
vector	O
spaces	O
over	O
the	O
real	O
numbers	O
Let	O
denote	O
the	O
real	O
part	O
of	O
and	O
observe	O
that	O
is	O
a	O
pairing	B-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
is	O
the	O
finest	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
for	O
which	O
this	O
is	O
true	O
,	O
which	O
is	O
what	O
makes	O
this	O
topology	O
important	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
topology	I-Algorithm
is	O
finer	O
than	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Throughout	O
this	O
section	O
,	O
will	O
be	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
with	O
continuous	O
dual	O
space	O
and	O
will	O
be	O
the	O
canonical	O
pairing	B-Algorithm
,	O
where	O
by	O
definition	O
The	O
vector	O
space	O
always	O
distinguishes/separates	O
the	O
points	O
of	O
but	O
may	O
fail	O
to	O
distinguishes	O
the	O
points	O
of	O
(	O
this	O
necessarily	O
happens	O
if	O
,	O
for	O
instance	O
,	O
is	O
not	O
Hausdorff	O
)	O
,	O
in	O
which	O
case	O
the	O
pairing	B-Algorithm
is	O
not	O
a	O
dual	B-Algorithm
pair	I-Algorithm
.	O
</s>
<s>
By	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
,	O
if	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
separates	O
points	O
of	O
and	O
thus	O
forms	O
a	O
dual	B-Algorithm
pair	I-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
and	O
is	O
a	O
collection	O
of	O
bounded	B-Algorithm
subsets	O
of	O
that	O
satisfies	O
axioms	O
and	O
If	O
contains	O
all	O
compact	O
subsets	O
of	O
then	O
is	O
complete	O
.	O
</s>
<s>
Throughout	O
,	O
will	O
be	O
a	O
TVS	O
over	O
the	O
field	O
with	O
continuous	O
dual	O
space	O
and	O
and	O
will	O
be	O
associated	O
with	O
the	O
canonical	O
pairing	B-Algorithm
.	O
</s>
<s>
Notation	O
:	O
If	O
denotes	O
a	O
polar	B-Algorithm
topology	I-Algorithm
then	O
endowed	O
with	O
this	O
topology	O
will	O
be	O
denoted	O
by	O
(	O
e.g.	O
</s>
<s>
The	O
reason	O
why	O
some	O
of	O
the	O
above	O
collections	O
(	O
in	O
the	O
same	O
row	O
)	O
induce	O
the	O
same	O
polar	B-Algorithm
topologies	I-Algorithm
is	O
due	O
to	O
some	O
basic	O
results	O
.	O
</s>
<s>
totally	O
bounded	B-Algorithm
)	O
subset	O
is	O
again	O
compact	O
(	O
resp	O
.	O
</s>
<s>
totally	O
bounded	B-Algorithm
)	O
.	O
</s>
<s>
If	O
is	O
bounded	B-Algorithm
then	O
is	O
absorbing	B-Algorithm
in	O
(	O
note	O
that	O
being	O
absorbing	B-Algorithm
is	O
a	O
necessary	O
condition	O
for	O
to	O
be	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
any	O
TVS	O
topology	O
on	O
)	O
.	O
</s>
<s>
Theorem	O
(	O
S	O
.	O
Banach	O
)	O
:	O
Suppose	O
that	O
and	O
are	O
Fréchet	B-Algorithm
spaces	I-Algorithm
or	O
that	O
they	O
are	O
duals	O
of	O
reflexive	O
Fréchet	B-Algorithm
spaces	I-Algorithm
and	O
that	O
is	O
a	O
continuous	O
linear	O
map	O
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
that	O
is	O
a	O
separately-continuous	O
bilinear	O
map	O
.	O
</s>
<s>
In	O
particular	O
,	O
any	O
separately	O
continuous	O
bilinear	O
maps	O
from	O
the	O
product	O
of	O
two	O
duals	O
of	O
reflexive	O
Fréchet	B-Algorithm
spaces	I-Algorithm
into	O
a	O
third	O
one	O
is	O
continuous	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
space	O
and	O
that	O
is	O
its	O
completion	O
.	O
</s>
<s>
Any	O
equicontinuous	O
subset	O
in	O
the	O
dual	O
of	O
a	O
separable	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
metrizable	O
in	O
the	O
topology	O
.	O
</s>
<s>
If	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
or	O
a	O
LF-space	B-Algorithm
then	O
is	O
complete	O
.	O
</s>
<s>
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
:	O
An	O
equicontinuous	O
subset	O
has	O
compact	O
closure	O
in	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
precompact	O
sets	O
.	O
</s>
<s>
If	O
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
(	O
e.g.	O
</s>
<s>
metrizable	O
or	O
LF-space	B-Algorithm
)	O
then	O
is	O
complete	O
.	O
</s>
<s>
If	O
is	O
a	O
LF-space	B-Algorithm
that	O
is	O
the	O
inductive	O
limit	O
of	O
the	O
sequence	O
of	O
space	O
(	O
for	O
)	O
then	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
all	O
are	O
normable	O
.	O
</s>
<s>
On	O
bounded	B-Algorithm
subsets	O
of	O
the	O
strong	O
and	O
weak	O
topologies	O
coincide	O
(	O
and	O
hence	O
so	O
do	O
all	O
other	O
topologies	O
finer	O
than	O
and	O
coarser	O
than	O
)	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
(	O
e.g.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
with	O
metrizable	O
and	O
that	O
is	O
a	O
linear	O
map	O
.	O
</s>
<s>
If	O
is	O
locally	B-Algorithm
convex	I-Algorithm
and	O
Hausdorff	O
then	O
'	O
s	O
given	O
topology	O
(	O
i.e.	O
</s>
<s>
That	O
is	O
,	O
for	O
Hausdorff	O
and	O
locally	B-Algorithm
convex	I-Algorithm
,	O
if	O
then	O
is	O
equicontinuous	O
if	O
and	O
only	O
if	O
is	O
equicontinuous	O
and	O
furthermore	O
,	O
for	O
any	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
if	O
and	O
only	O
if	O
is	O
equicontinuous	O
.	O
</s>
<s>
Importantly	O
,	O
a	O
set	O
of	O
continuous	O
linear	O
functionals	O
on	O
a	O
TVS	O
is	O
equicontinuous	O
if	O
and	O
only	O
if	O
it	O
is	O
contained	O
in	O
the	O
polar	B-Algorithm
of	O
some	O
neighborhood	O
of	O
the	O
origin	O
in	O
(	O
i.e.	O
</s>
<s>
Since	O
a	O
TVS	O
's	O
topology	O
is	O
completely	O
determined	O
by	O
the	O
open	O
neighborhoods	O
of	O
the	O
origin	O
,	O
this	O
means	O
that	O
via	O
operation	O
of	O
taking	O
the	O
polar	B-Algorithm
of	O
a	O
set	O
,	O
the	O
collection	O
of	O
equicontinuous	O
subsets	O
of	O
"	O
encode	O
"	O
all	O
information	O
about	O
'	O
s	O
topology	O
(	O
i.e.	O
</s>
<s>
Thus	O
uniform	B-Algorithm
convergence	I-Algorithm
on	O
the	O
collection	O
of	O
equicontinuous	O
subsets	O
is	O
essentially	O
"	O
convergence	O
on	O
the	O
topology	O
of	O
"	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
space	O
.	O
</s>
<s>
Let	O
be	O
a	O
vector	O
space	O
and	O
let	O
be	O
a	O
vector	O
subspace	O
of	O
the	O
algebraic	O
dual	O
of	O
that	O
separates	O
points	O
on	O
If	O
is	O
any	O
other	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
topology	O
on	O
then	O
is	O
compatible	O
with	O
duality	O
between	O
and	O
if	O
when	O
is	O
equipped	O
with	O
then	O
it	O
has	O
as	O
its	O
continuous	O
dual	O
space	O
.	O
</s>
<s>
If	O
is	O
given	O
the	O
weak	O
topology	O
then	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	O
)	O
and	O
is	O
compatible	O
with	O
duality	O
between	O
and	O
(	O
i.e.	O
</s>
<s>
The	O
question	O
arises	O
:	O
what	O
are	O
all	O
of	O
the	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
TVS	O
topologies	O
that	O
can	O
be	O
placed	O
on	O
that	O
are	O
compatible	O
with	O
duality	O
between	O
and	O
?	O
</s>
<s>
The	O
answer	O
to	O
this	O
question	O
is	O
called	O
the	O
Mackey	B-Algorithm
–	I-Algorithm
Arens	I-Algorithm
theorem	I-Algorithm
.	O
</s>
