<s>
In	O
each	O
case	O
,	O
the	O
definition	O
describes	O
a	O
duality	O
between	O
certain	O
subsets	O
of	O
a	O
pairing	B-Algorithm
of	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
over	O
the	O
real	O
or	O
complex	O
numbers	O
(	O
and	O
are	O
often	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
then	O
the	O
space	O
will	O
usually	O
,	O
but	O
not	O
always	O
,	O
be	O
the	O
continuous	O
dual	O
space	O
of	O
in	O
which	O
case	O
the	O
dual	O
pairing	B-Algorithm
will	O
again	O
be	O
the	O
evaluation	O
map	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
pairing	B-Algorithm
.	O
</s>
<s>
This	O
is	O
an	O
affine	B-Algorithm
shift	I-Algorithm
of	O
the	O
geometric	O
definition	O
;	O
</s>
<s>
it	O
has	O
the	O
useful	O
characterization	O
that	O
the	O
functional-analytic	O
polar	O
of	O
the	O
unit	O
ball	O
(	O
in	O
)	O
is	O
precisely	O
the	O
unit	O
ball	O
(	O
in	O
)	O
.	O
</s>
<s>
Thus	O
for	O
all	O
of	O
these	O
definitions	O
of	O
the	O
polar	B-Algorithm
set	I-Algorithm
of	O
to	O
agree	O
,	O
it	O
suffices	O
that	O
for	O
all	O
scalars	O
of	O
unit	O
length	O
(	O
where	O
this	O
is	O
equivalent	O
to	O
for	O
all	O
unit	O
length	O
scalar	O
)	O
.	O
</s>
<s>
If	O
is	O
any	O
vector	O
space	O
then	O
let	O
denote	O
the	O
algebraic	O
dual	O
space	O
of	O
which	O
is	O
the	O
set	O
of	O
all	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
The	O
vector	O
space	O
is	O
always	O
a	O
closed	O
subset	O
of	O
the	O
space	O
of	O
all	O
-valued	O
functions	O
on	O
under	O
the	O
topology	O
of	O
pointwise	O
convergence	O
so	O
when	O
is	O
endowed	O
with	O
the	O
subspace	O
topology	O
,	O
then	O
becomes	O
a	O
Hausdorff	O
complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	O
)	O
.	O
</s>
<s>
For	O
any	O
finite-dimensional	O
vector	O
subspace	O
of	O
let	O
denote	O
the	O
Euclidean	O
topology	O
on	O
which	O
is	O
the	O
unique	O
topology	O
that	O
makes	O
into	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
.	O
</s>
<s>
The	O
polar	O
of	O
a	O
subset	O
with	O
respect	O
to	O
this	O
canonical	O
pairing	B-Algorithm
is	O
:	O
</s>
<s>
The	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
states	O
that	O
if	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
then	O
and	O
this	O
polar	B-Algorithm
set	I-Algorithm
is	O
a	O
compact	O
subset	O
of	O
the	O
continuous	O
dual	O
space	O
when	O
is	O
endowed	O
with	O
the	O
weak-*	O
topology	O
(	O
also	O
known	O
as	O
the	O
topology	O
of	O
pointwise	O
convergence	O
)	O
.	O
</s>
<s>
The	O
bipolar	O
theorem	O
characterizes	O
the	O
bipolar	O
of	O
a	O
subset	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
.	O
</s>
<s>
Unless	O
stated	O
otherwise	O
,	O
will	O
be	O
a	O
pairing	B-Algorithm
.	O
</s>
<s>
is	O
closed	O
in	O
under	O
the	O
weak-*	B-Algorithm
-topology	I-Algorithm
on	O
.	O
</s>
<s>
-bounded	O
)	O
if	O
and	O
only	O
if	O
is	O
absorbing	B-Algorithm
in	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
then	O
the	O
polars	O
(	O
taken	O
with	O
respect	O
to	O
)	O
of	O
any	O
0-neighborhood	O
base	O
forms	O
a	O
fundamental	O
family	O
of	O
equicontinuous	O
subsets	O
of	O
(	O
i.e.	O
</s>
<s>
The	O
last	O
two	O
results	O
explain	O
why	O
equicontinuous	O
subsets	O
of	O
the	O
continuous	O
dual	O
space	O
play	O
such	O
a	O
prominent	O
role	O
in	O
the	O
modern	O
theory	O
of	O
functional	B-Application
analysis	I-Application
:	O
because	O
equicontinuous	O
subsets	O
encapsulate	O
all	O
information	O
about	O
the	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
'	O
s	O
original	O
topology	O
.	O
</s>
