In functional analysis and related areas of mathematics the polar set of a given subset of a vector space is a certain set in the dual space.
Given a dual pair
(
X
,
Y
)
the polar set or polar of a subset
A
of
X
is a set
A
∘
in
Y
defined as
A
∘
:=
{
y
∈
Y
:
sup
x
∈
A
|
⟨
x
,
y
⟩
|
≤
1
}
The bipolar of a subset
A
of
X
is the polar of
A
∘
. It is denoted
A
∘
∘
and is a set in
X
.
A
∘
is absolutely convex
If
A
⊆
B
then
B
∘
⊆
A
∘
So
⋃
i
∈
I
A
i
∘
⊆
(
⋂
i
∈
I
A
i
)
∘
, where equality of sets does not necessarily hold.
For all
γ
≠
0
:
(
γ
A
)
∘
=
1
∣
γ
∣
A
∘
(
⋃
i
∈
I
A
i
)
∘
=
⋂
i
∈
I
A
i
∘
For a dual pair
(
X
,
Y
)
A
∘
is closed in
Y
under the weak-*-topology on
Y
The bipolar
A
∘
∘
of a set
A
is the absolutely convex envelope of
A
, that is the smallest absolutely convex set containing
A
. If
A
is already absolutely convex then
A
∘
∘
=
A
.
For a closed convex cone
C
in
X
, the polar cone is equivalent to the one-sided polar set for
C
, given by
C
∘
=
{
y
∈
Y
:
sup
{
⟨
x
,
y
⟩
:
x
∈
C
}
≤
1
}
.
In geometry, the polar set may also refer to a duality between points and planes. In particular, the polar set of a point
x
0
, given by the set of points
x
satisfying
⟨
x
,
x
0
⟩
=
0
is its polar hyperplane, and the dual relationship for a hyperplane yields its pole.
