<s>
In	O
mathematics	O
,	O
a	O
dual	B-Algorithm
system	I-Algorithm
,	O
dual	B-Algorithm
pair	I-Algorithm
,	O
or	O
duality	O
over	O
a	O
field	O
is	O
a	O
triple	O
consisting	O
of	O
two	O
vector	O
spaces	O
and	O
over	O
and	O
a	O
non-degenerate	B-Algorithm
bilinear	O
map	O
.	O
</s>
<s>
Duality	B-Algorithm
theory	I-Algorithm
,	O
the	O
study	O
of	O
dual	B-Algorithm
systems	I-Algorithm
,	O
is	O
part	O
of	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
According	O
to	O
Helmut	O
H	O
.	O
Schaefer	O
,	O
"	O
the	O
study	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
in	O
terms	O
of	O
its	O
dual	O
is	O
the	O
central	O
part	O
of	O
the	O
modern	O
theory	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
for	O
it	O
provides	O
the	O
deepest	O
and	O
most	O
beautiful	O
results	O
of	O
the	O
subject.	O
"	O
</s>
<s>
A	O
pairing	O
is	O
called	O
a	O
,	O
a	O
,	O
or	O
a	O
over	O
if	O
the	O
bilinear	O
form	O
is	O
non-degenerate	B-Algorithm
,	O
which	O
means	O
that	O
it	O
satisfies	O
the	O
following	O
two	O
separation	O
axioms	O
:	O
</s>
<s>
The	O
absolute	B-Algorithm
polar	I-Algorithm
or	O
polar	B-Algorithm
of	O
a	O
subset	O
of	O
is	O
the	O
set	O
:	O
</s>
<s>
the	O
definition	O
of	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
on	O
)	O
then	O
by	O
switching	O
the	O
order	O
of	O
and	O
then	O
it	O
is	O
meant	O
that	O
definition	O
applied	O
to	O
(	O
e.g.	O
</s>
<s>
The	O
following	O
notation	O
is	O
now	O
nearly	O
ubiquitous	O
in	O
duality	B-Algorithm
theory	I-Algorithm
.	O
</s>
<s>
If	O
separates	O
points	O
of	O
(	O
which	O
is	O
true	O
if	O
,	O
for	O
instance	O
,	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
)	O
then	O
this	O
pairing	O
forms	O
a	O
duality	O
.	O
</s>
<s>
The	O
following	O
result	O
shows	O
that	O
the	O
continuous	O
linear	O
functionals	O
on	O
a	O
TVS	O
are	O
exactly	O
those	O
linear	O
functionals	O
that	O
are	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
pre-Hilbert	O
space	O
is	O
a	O
dual	O
pairing	O
if	O
and	O
only	O
if	O
is	O
vector	O
space	O
over	O
or	O
has	O
dimension	O
Here	O
it	O
is	O
assumed	O
that	O
the	O
sesquilinear	B-Algorithm
form	I-Algorithm
is	O
conjugate	O
homogeneous	O
in	O
its	O
second	O
coordinate	O
and	O
homogeneous	O
in	O
its	O
first	O
coordinate	O
.	O
</s>
<s>
If	O
is	O
a	O
Hilbert	O
space	O
then	O
forms	O
a	O
dual	B-Algorithm
system	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
complex	O
Hilbert	O
space	O
then	O
forms	O
a	O
dual	B-Algorithm
system	I-Algorithm
if	O
and	O
only	O
if	O
If	O
is	O
non-trivial	O
then	O
does	O
not	O
even	O
form	O
pairing	O
since	O
the	O
inner	O
product	O
is	O
sesquilinear	B-Algorithm
rather	O
than	O
bilinear	O
.	O
</s>
<s>
Let	O
(	O
where	O
is	O
such	O
that	O
)	O
,	O
and	O
Then	O
is	O
a	O
dual	B-Algorithm
system	I-Algorithm
.	O
</s>
<s>
A	O
sequence	B-Algorithm
space	I-Algorithm
and	O
its	O
beta	B-Algorithm
dual	I-Algorithm
with	O
the	O
bilinear	O
map	O
defined	O
as	O
for	O
forms	O
a	O
dual	B-Algorithm
system	I-Algorithm
.	O
</s>
<s>
-converges	O
,	O
-bounded	O
,	O
etc	O
.	O
)	O
</s>
<s>
If	O
is	O
a	O
sequence	O
of	O
orthonormal	B-Algorithm
vectors	I-Algorithm
in	O
Hilbert	O
space	O
,	O
then	O
converges	O
weakly	O
to	O
0	O
but	O
does	O
not	O
norm-converge	O
to	O
0	O
(	O
or	O
any	O
other	O
vector	O
)	O
.	O
</s>
<s>
which	O
(	O
among	O
other	O
things	O
)	O
allows	O
for	O
to	O
be	O
endowed	O
with	O
the	O
subspace	O
topology	O
induced	O
on	O
it	O
by	O
,	O
say	O
,	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
(	O
this	O
topology	O
is	O
also	O
called	O
the	O
strong	O
bidual	O
topology	O
and	O
it	O
appears	O
in	O
the	O
theory	O
of	O
reflexive	O
spaces	O
:	O
the	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
said	O
to	O
be	O
if	O
and	O
it	O
will	O
be	O
called	O
if	O
in	O
addition	O
the	O
strong	O
bidual	O
topology	O
on	O
is	O
equal	O
to	O
'	O
s	O
original/starting	O
topology	O
)	O
.	O
</s>
<s>
The	O
following	O
results	O
are	O
important	O
for	O
defining	O
polar	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
If	O
in	O
addition	O
distinguishes	O
points	O
of	O
then	O
is	O
-bounded	O
if	O
and	O
only	O
if	O
it	O
is	O
-totally	O
bounded	B-Algorithm
.	O
</s>
<s>
Let	O
and	O
let	O
denotes	O
the	O
absolute	B-Algorithm
polar	I-Algorithm
of	O
then	O
:	O
</s>
<s>
These	O
results	O
hold	O
when	O
the	O
real	O
polar	B-Algorithm
is	O
used	O
in	O
place	O
of	O
the	O
absolute	B-Algorithm
polar	I-Algorithm
.	O
</s>
<s>
Then	O
every	O
-bounded	O
subset	O
of	O
is	O
contained	O
in	O
a	O
finite	O
dimensional	O
vector	O
subspace	O
and	O
every	O
vector	O
subspace	O
of	O
is	O
-closed	O
.	O
</s>
<s>
If	O
is	O
a	O
complete	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
say	O
that	O
is	O
-complete	O
or	O
(	O
if	O
no	O
ambiguity	O
can	O
arise	O
)	O
weakly-complete	O
.	O
</s>
<s>
There	O
exist	O
Banach	O
spaces	O
that	O
are	O
not	O
weakly-complete	O
(	O
despite	O
being	O
complete	B-Algorithm
in	O
their	O
norm	O
topology	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
vector	O
space	O
then	O
under	O
the	O
canonical	O
duality	O
,	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
Said	O
differently	O
,	O
there	O
does	O
exist	O
a	O
proper	O
vector	O
subspace	O
of	O
such	O
that	O
is	O
Hausdorff	O
and	O
is	O
complete	B-Algorithm
in	O
the	O
weak-*	O
topology	O
(	O
i.e.	O
</s>
<s>
Consequently	O
,	O
when	O
the	O
continuous	O
dual	O
space	O
of	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
endowed	O
with	O
the	O
weak-*	O
topology	O
,	O
then	O
is	O
complete	B-Algorithm
if	O
and	O
only	O
if	O
(	O
that	O
is	O
,	O
if	O
and	O
only	O
if	O
linear	O
functional	O
on	O
is	O
continuous	O
)	O
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
canonical	O
pairings	O
(	O
so	O
and	O
)	O
that	O
are	O
dual	B-Algorithm
systems	I-Algorithm
and	O
let	O
be	O
a	O
linear	O
map	O
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
dual	B-Algorithm
systems	I-Algorithm
and	O
is	O
a	O
weakly	O
continuous	O
linear	O
map	O
.	O
</s>
<s>
If	O
is	O
a	O
linear	O
map	O
between	O
two	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
then	O
:	O
</s>
<s>
If	O
is	O
separable	O
and	O
is	O
equicontinuous	O
then	O
when	O
endowed	O
with	O
the	O
subspace	O
topology	O
induced	O
by	O
is	O
metrizable	B-Algorithm
.	O
</s>
<s>
If	O
is	O
separable	O
and	O
metrizable	B-Algorithm
,	O
then	O
is	O
separable	O
.	O
</s>
<s>
Starting	O
with	O
only	O
the	O
weak	O
topology	O
,	O
the	O
use	O
of	O
polar	B-Algorithm
sets	I-Algorithm
produces	O
a	O
range	O
of	O
locally	B-Algorithm
convex	I-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
Such	O
topologies	O
are	O
called	O
polar	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
Every	O
polar	B-Algorithm
topology	I-Algorithm
is	O
necessarily	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
The	O
following	O
table	O
lists	O
some	O
of	O
the	O
more	O
important	O
polar	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
:	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>
A	O
subset	O
of	O
is	O
weakly	O
bounded	B-Algorithm
(	O
resp	O
.	O
</s>
<s>
Mackey	O
bounded	B-Algorithm
,	O
strongly	O
bounded	B-Algorithm
)	O
if	O
it	O
is	O
bounded	B-Algorithm
in	O
(	O
resp	O
.	O
</s>
<s>
bounded	B-Algorithm
in	O
bounded	B-Algorithm
in	O
)	O
.	O
</s>
<s>
There	O
is	O
a	O
strongest	O
topology	O
compatible	O
with	O
this	O
pairing	O
and	O
that	O
is	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
The	O
following	O
is	O
one	O
of	O
the	O
most	O
important	O
theorems	O
in	O
duality	B-Algorithm
theory	I-Algorithm
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
whose	O
given	O
topology	O
is	O
identical	O
to	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
is	O
called	O
a	O
Mackey	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
above	O
theorem	O
implies	O
that	O
the	O
closed	O
and	O
convex	O
subsets	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
depend	O
on	O
the	O
continuous	O
dual	O
space	O
.	O
</s>
<s>
that	O
is	O
,	O
if	O
and	O
are	O
any	O
locally	B-Algorithm
convex	I-Algorithm
topologies	I-Algorithm
on	O
with	O
the	O
same	O
continuous	O
dual	O
spaces	O
,	O
then	O
a	O
convex	O
subset	O
of	O
is	O
closed	O
in	O
the	O
topology	O
if	O
and	O
only	O
if	O
it	O
is	O
closed	O
in	O
the	O
topology	O
.	O
</s>
<s>
The	O
following	O
theorem	O
shows	O
that	O
barrels	B-Algorithm
(	O
i.e.	O
</s>
<s>
closed	O
absorbing	B-Algorithm
disks	O
)	O
are	O
exactly	O
the	O
polars	O
of	O
weakly	O
bounded	B-Algorithm
subsets	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
then	O
:	O
</s>
<s>
A	O
closed	O
absorbing	B-Algorithm
and	O
balanced	O
subset	O
of	O
absorbs	O
each	O
convex	O
compact	O
subset	O
of	O
(	O
i.e.	O
</s>
<s>
All	O
of	O
this	O
leads	O
to	O
Mackey	O
's	O
theorem	O
,	O
which	O
is	O
one	O
of	O
the	O
central	O
theorems	O
in	O
the	O
theory	O
of	O
dual	B-Algorithm
systems	I-Algorithm
.	O
</s>
<s>
In	O
short	O
,	O
it	O
states	O
the	O
bounded	B-Algorithm
subsets	O
are	O
the	O
same	O
for	O
any	O
two	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
topologies	I-Algorithm
that	O
are	O
compatible	O
with	O
the	O
same	O
duality	O
.	O
</s>
<s>
Moreover	O
,	O
a	O
subset	O
is	O
-bounded	O
(	O
resp	O
.	O
</s>
<s>
-bounded	O
)	O
if	O
and	O
only	O
if	O
there	O
exists	O
a	O
sequence	O
of	O
positive	O
real	O
numbers	O
such	O
that	O
for	O
all	O
and	O
all	O
indices	O
(	O
resp	O
.	O
</s>
<s>
It	O
follows	O
that	O
there	O
are	O
weakly	O
bounded	B-Algorithm
(	O
that	O
is	O
,	O
-bounded	O
)	O
subsets	O
of	O
that	O
are	O
not	O
strongly	O
bounded	B-Algorithm
(	O
that	O
is	O
,	O
not	O
-bounded	O
)	O
.	O
</s>
