In functional analysis and related areas of mathematics a polar topology, topology of
A
-convergence or topology of uniform convergence on the sets of
A
is a method to define locally convex topologies on the vector spaces of a dual pair.
Let
(
X
,
Y
,
⟨
,
⟩
)
be a dual pair of vector spaces
X
and
Y
over the field
F
, either the real or complex numbers.
A set
A
⊆
X
is said to be bounded in
X
with respect to
Y
, if for each element
y
∈
Y
the set of values
{
⟨
x
,
y
⟩
;
x
∈
A
}
is bounded:
∀
y
∈
Y
sup
x
∈
A
|
⟨
x
,
y
⟩
|
<
∞
.
This condition is equivalent to the requirement that the polar
A
∘
of the set
A
in
Y
A
∘
=
{
y
∈
Y
:
sup
x
∈
A
|
⟨
x
,
y
⟩
|
≤
1
}
is an absorbent set in
Y
, i.e.
⋃
λ
∈
F
λ
⋅
A
∘
=
Y
.
Let now
A
be a family of bounded sets in
X
(with respect to
Y
) with the following properties:
each point
x
of
X
belongs to some set
A
∈
A
∀
x
∈
X
∃
A
∈
A
x
∈
A
,
each two sets
A
∈
A
and
B
∈
A
are contained in some set
C
∈
A
:
∀
A
,
B
∈
A
∃
C
∈
A
A
∪
B
⊆
C
,
A
is closed under the operation of multiplication by scalars:
∀
A
∈
A
∀
λ
∈
F
λ
⋅
A
∈
A
.
Then the seminorms of the form
∥
y
∥
A
=
sup
x
∈
A
|
⟨
x
,
y
⟩
|
,
A
∈
A
,
define a Hausdorff locally convex topology on
Y
which is called the polar topology on
Y
generated by the family of sets
A
. The sets
U
A
=
{
x
∈
V
:
∥
φ
∥
A
<
1
}
,
A
∈
A
,
form a local base of this topology. A net of elements
y
i
∈
Y
tends to an element
y
∈
Y
in this topology if and only if
∀
A
∈
A
∥
y
i
−
y
∥
A
=
sup
x
∈
A
|
⟨
x
,
y
i
⟩
−
⟨
x
,
y
⟩
|
⟶
i
→
∞
0.
Because of this the polar topology is often called the topology of uniform convergence on the sets of
A
. The semi norm
∥
y
∥
A
is the gauge of the polar set
A
∘
.
if
A
is the family of all bounded sets in
X
then the polar topology on
Y
coincides with the strong topology,
if
A
is the family of all finite sets in
X
then the polar topology on
Y
coincides with the weak topology,
the topology of an arbitrary locally convex space
X
can be described as the polar topology defined on
X
by the family
A
of all equicontinuous sets
A
⊆
X
′
in the dual space
X
′
.
