<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
a	O
dual	B-Algorithm
topology	I-Algorithm
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
a	O
vector	O
space	O
that	O
is	O
induced	O
by	O
the	O
continuous	O
dual	O
of	O
the	O
vector	O
space	O
,	O
by	O
means	O
of	O
the	O
bilinear	O
form	O
(	O
also	O
called	O
pairing	O
)	O
associated	O
with	O
the	O
dual	B-Algorithm
pair	I-Algorithm
.	O
</s>
<s>
The	O
different	O
dual	B-Algorithm
topologies	I-Algorithm
for	O
a	O
given	O
dual	B-Algorithm
pair	I-Algorithm
are	O
characterized	O
by	O
the	O
Mackey	O
–	O
Arens	O
theorem	O
.	O
</s>
<s>
All	O
locally	B-Algorithm
convex	I-Algorithm
topologies	I-Algorithm
with	O
their	O
continuous	O
dual	O
are	O
trivially	O
a	O
dual	B-Algorithm
pair	I-Algorithm
and	O
the	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
is	O
a	O
dual	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Several	O
topological	O
properties	O
depend	O
only	O
on	O
the	O
dual	B-Algorithm
pair	I-Algorithm
and	O
not	O
on	O
the	O
chosen	O
dual	B-Algorithm
topology	I-Algorithm
and	O
thus	O
it	O
is	O
often	O
possible	O
to	O
substitute	O
a	O
complicated	O
dual	B-Algorithm
topology	I-Algorithm
by	O
a	O
simpler	O
one	O
.	O
</s>
<s>
(	O
If	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
is	O
not	O
a	O
dual	B-Algorithm
topology	I-Algorithm
,	O
then	O
either	O
is	O
not	O
surjective	O
or	O
it	O
is	O
ill-defined	O
since	O
the	O
linear	O
functional	O
is	O
not	O
continuous	O
on	O
for	O
some	O
.	O
)	O
</s>
<s>
Theorem	O
(	O
by	O
Mackey	O
)	O
:	O
Given	O
a	O
dual	B-Algorithm
pair	I-Algorithm
,	O
the	O
bounded	B-Algorithm
sets	I-Algorithm
under	O
any	O
dual	B-Algorithm
topology	I-Algorithm
are	O
identical	O
.	O
</s>
<s>
Under	O
any	O
dual	B-Algorithm
topology	I-Algorithm
the	O
same	O
sets	O
are	O
barrelled	B-Algorithm
.	O
</s>
<s>
The	O
Mackey	O
–	O
Arens	O
theorem	O
,	O
named	O
after	O
George	O
Mackey	O
and	O
Richard	O
Arens	O
,	O
characterizes	O
all	O
possible	O
dual	B-Algorithm
topologies	I-Algorithm
on	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
theorem	O
shows	O
that	O
the	O
coarsest	O
dual	B-Algorithm
topology	I-Algorithm
is	O
the	O
weak	O
topology	O
,	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
all	O
finite	O
subsets	O
of	O
,	O
and	O
the	O
finest	O
topology	O
is	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
,	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
all	O
absolutely	O
convex	O
weakly	O
compact	O
subsets	O
of	O
.	O
</s>
