In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology.
Let
(
X
,
Y
,
⟨
,
⟩
)
be a dual pair of vector spaces over the field
F
of real (
R
) or complex (
C
) numbers. Let us denote by
B
the system of all subsets
B
⊆
X
bounded by elements of
Y
in the following sense:
∀
y
∈
Y
sup
x
∈
B
|
⟨
x
,
y
⟩
|
<
∞
.
Then the strong topology
β
(
Y
,
X
)
on
Y
is defined as the locally convex topology on
Y
generated by the seminorms of the form
|
|
y
|
|
B
=
sup
x
∈
B
|
⟨
x
,
y
⟩
|
,
y
∈
Y
,
B
∈
B
.
In the special case when
X
is a locally convex space, the strong topology on the (continuous) dual space
X
′
(i.e. on the space of all continuous linear functionals
f
:
X
→
F
) is defined as the strong topology
β
(
X
′
,
X
)
, and it coincides with the topology of uniform convergence on bounded sets in
X
, i.e. with the topology on
X
′
generated by the seminorms of the form
|
|
f
|
|
B
=
sup
x
∈
B
|
f
(
x
)
|
,
f
∈
X
′
,
where
B
runs over the family of all bounded sets in
X
. The space
X
′
with this topology is called strong dual space of the space
X
and is denoted by
X
β
′
.
If
X
is a normed vector space, then its (continuous) dual space
X
′
with the strong topology coincides with the Banach dual space
X
′
, i.e. with the space
X
′
with the topology induced by the operator norm. Conversely
β
(
X
,
X
′
)
-topology on
X
is identical to the topology induced by the norm on
X
.
If
X
is a barrelled space, then its topology coincides with the strong topology
β
(
X
,
X
′
)
on
X
and with the Mackey topology on
X
generated by the pairing
(
X
,
X
′
)
.
