Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups:
Suitably regular complexvalued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series;
Suitably regular complexvalued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms; and
Complexvalued functions on a finite abelian group have discrete Fourier transforms which are functions on the dual group, which is a (noncanonically) isomorphic group. Moreover, any function on a finite group can be recovered from its discrete Fourier transform.
The theory, introduced by Lev Pontryagin and combined with Haar measure introduced by John von Neumann, André Weil and others depends on the theory of the dual group of a locally compact abelian group.
It is analogous to the dual vector space of a vector space: a finitedimensional vector space V and its dual vector space V* are not naturally isomorphic, but their endomorphism algebras (matrix algebras) are:
End
(
V
)
≅
End
(
V
∗
)
,
via the transpose. Similarly, a group G and its dual group
G
^
are not in general isomorphic, but their group algebras are:
C
(
G
)
≅
C
(
G
^
)
via the Fourier transform, though one must carefully define these algebras analytically. More categorically, this is not just an isomorphism of endomorphism algebras, but an isomorphism of categories – see categorical considerations.
A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff. It's called abelian if the underlying group is abelian.
Examples of locally compact abelian groups are:
R
n
for n a positive integer, with vector addition as group operation.
The positive real numbers
R
+
with multiplication as operation. This group is isomorphic to
(
R
,
+
)
by the exponential map.
Any finite abelian group, with the discrete topology. By the structure theorem for finite abelian groups, all such groups are products of cyclic groups.
The integers
Z
under addition, again with the discrete topology.
The circle group, denoted
T
for torus. This is the group of complex numbers of modulus 1.
T
is isomorphic as a topological group to the quotient group
R
/
Z
.
The field
Q
p
of padic numbers under addition, with the usual padic topology.
If
G
is a locally compact abelian group, a character of
G
is a continuous group homomorphism from
G
with values in the circle group
T
. The set of all characters on
G
can be made into a locally compact abelian group, called the dual group of
G
and denoted
G
^
. The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets (i.e., the compactopen topology, viewing
G
^
as a subset of the space of all continuous functions from
G
to
T
.). This topology in general is not metrizable. However, if the group
G
is a separable locally compact abelian group, then the dual group is metrizable.
This is analogous to the dual space in linear algebra: just as for a vector space
V
over a field
K
, the dual space is
H
o
m
(
V
,
K
)
, so too is the dual group
H
o
m
(
G
,
T
)
. More abstractly, these are both examples of representable functors, being represented respectively by
K
and
T
.
A group that is isomorphic (as topological groups) to its dual group is called selfdual. While the reals and finite cyclic groups are selfdual, the group and the dual group are not naturally isomorphic, and should be thought of as two different groups.
The dual of
Z
is isomorphic to the circle group
T
. A character on the infinite cyclic group of integers
Z
under addition is determined by its value at the generator 1. Thus for any character
χ
on
Z
,
χ
(
n
)
=
χ
(
1
)
n
. Moreover, this formula defines a character for any choice of
χ
(
1
)
in
T
. The topology of uniform convergence on compact sets is in this case the topology of pointwise convergence. This is the topology of the circle group inherited from the complex numbers.
The dual of
T
is canonically isomorphic with
Z
. Indeed, a character on
T
is of the form
z
↦
z
n
for
n
an integer. Since
T
is compact, the topology on the dual group is that of uniform convergence, which turns out to be the discrete topology.
The group of real numbers
R
, is isomorphic to its own dual; the characters on
R
are of the form
r
↦
e
i
θ
r
for
θ
a real number. With these dualities, the version of the Fourier transform to be introduced next coincides with the classical Fourier transform on
R
.
Analogously, the group of
p
adic numbers
Q
p
is isomorphic to its dual. (In fact, any finite extension of
Q
p
is also selfdual.) It follows that the adeles are selfdual.
Theorem. There is a canonical isomorphism
G
≅
G
^
^
between any locally compact abelian group
G
and its double dual.
Canonical means that there is a naturally defined map
e
v
G
:
G
→
G
^
^
; more importantly, the map should be functorial in
G
. The canonical isomorphism is defined on
x
∈
G
as follows:
e
v
G
(
x
)
=
{
χ
↦
χ
(
x
)
}
i.e.
e
v
G
(
x
)
(
χ
)
:=
χ
(
x
)
∈
T
.
In other words, each group element
x
is identified to the evaluation character on the dual. This is strongly analogous to the canonical isomorphism between a finitedimensional vector space and its double dual,
V
≅
V
∗
∗
. However, there is also a difference:
V
is isomorphic to its dual space
V
∗
, although not canonically so, while many groups are not isomorphic to their dual groups (for instance
T
^
=
Z
but
T
≇
Z
as topological groups). If
G
is a finite abelian group, then
G
≅
G
^
but this isomorphism is not canonical. Making this statement precise (in general) requires thinking about dualizing not only on groups, but also on maps between the groups, in order to treat dualization as a functor and prove the identity functor and the dualization functor are not naturally equivalent. Also it should be noted that the duality theorem implies that for any group (not necessarily finite) the dualization functor is an exact functor.
One of the most remarkable facts about a locally compact group G is that it carries an essentially unique natural measure, the Haar measure, which allows one to consistently measure the "size" of sufficiently regular subsets of G. "Sufficiently regular subset" here means a Borel set; that is, an element of the σalgebra generated by the compact sets. More precisely, a right Haar measure on a locally compact group G is a countably additive measure μ defined on the Borel sets of G which is right invariant in the sense that μ(Ax) = μ(A) for x an element of G and A a Borel subset of G and also satisfies some regularity conditions (spelled out in detail in the article on Haar measure). Except for positive scaling factors, a Haar measure on G is unique.
The Haar measure on G allows us to define the notion of integral for (complexvalued) Borel functions defined on the group. In particular, one may consider various L^{p} spaces associated to the Haar measure μ. Specifically,
L
μ
p
(
G
)
=
{
f
:
G
→
C
:
∫
G

f
(
x
)

p
d
μ
(
x
)
<
∞
}
.
Note that, since any two Haar measures on G are equal up to a scaling factor, this L^{p}space is independent of the choice of Haar measure and thus perhaps could be written as L^{p}(G). However, the L^{p}norm on this space depends on the choice of Haar measure, so if one wants to talk about isometries it is important to keep track of the Haar measure being used.
The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform. If
f
∈
L
1
(
G
)
, then the Fourier transform is the function
f
^
on
G
^
defined by
f
^
(
χ
)
=
∫
G
f
(
x
)
χ
(
x
)
¯
d
μ
(
x
)
,
where the integral is relative to Haar measure
μ
on
G
. This is also denoted
(
F
f
)
(
χ
)
. Note the Fourier transform depends on the choice of Haar measure. It is not too difficult to show that the Fourier transform of an
L
1
function on
G
is a bounded continuous function on
G
^
which vanishes at infinity.
Fourier Inversion Formula for
L
1
Functions. For each Haar measure
μ
on
G
there is a unique Haar measure
ν
on
G
^
such that whenever
f
∈
L
1
(
G
)
and
f
^
∈
L
1
(
G
^
)
, we have
f
(
x
)
=
∫
G
^
f
^
(
χ
)
χ
(
x
)
d
ν
(
χ
)
μ
almost everywhere
If
f
is continuous then this identity holds for all
x
.
The inverse Fourier transform of an integrable function on
G
^
is given by
g
ˇ
(
x
)
=
∫
G
^
g
(
χ
)
χ
(
x
)
d
ν
(
χ
)
,
where the integral is relative to the Haar measure
ν
on the dual group
G
^
. The measure
ν
on
G
^
that appears in the Fourier inversion formula is called the dual measure to
μ
and may be denoted
μ
^
.
The various Fourier transforms can be classified in terms of their domain and transform domain (the group and dual group) as follows:
As an example, suppose
G
=
R
n
, so we can think about
G
^
as
R
n
by the pairing
(
v
,
w
)
↦
e
i
v
⋅
w
.
If
μ
is the Lebesgue measure on Euclidean space, we obtain the ordinary Fourier transform on
R
n
and the dual measure needed for the Fourier inversion formula is
μ
^
=
(
2
π
)
−
n
μ
. If we want to get a Fourier inversion formula with the same measure on both sides (that is, since we can think about
R
n
as its own dual space we can ask for
μ
^
to equal
μ
) then we need to use
μ
=
(
2
π
)
−
n
2
×
Lebesgue measure
μ
^
=
(
2
π
)
−
n
2
×
Lebesgue measure
However, if we change the way we identify
R
n
with its dual group, by using the pairing
(
v
,
w
)
↦
e
2
π
i
v
⋅
w
,
then Lebesgue measure on
R
n
is equal to its own dual measure. This convention minimizes the number of factors of
2
π
that show up in various places when computing Fourier transforms or inverse Fourier transforms on Euclidean space. (In effect it limits the
2
π
only to the exponent rather than as some messy factor outside the integral sign.) Note that the choice of how to identify
R
n
with its dual group affects the meaning of the term "selfdual function", which is a function on
R
n
equal to its own Fourier transform: using the classical pairing
(
v
,
w
)
↦
e
i
v
⋅
w
the function
e
−
1
2
x
2
is selfdual, but using the (cleaner) pairing
(
v
,
w
)
↦
e
2
π
i
v
⋅
w
makes
e
−
π
x
2
selfdual instead.
The space of integrable functions on a locally compact abelian group G is an algebra, where multiplication is convolution: the convolution of two integrable functions f and g is defined as
(
f
∗
g
)
(
x
)
=
∫
G
f
(
x
−
y
)
g
(
y
)
d
μ
(
y
)
.
Theorem. The Banach space
L
1
(
G
)
is an associative and commutative algebra under convolution.
This algebra is referred to as the Group Algebra of G. By the Fubini–Tonelli theorem, the convolution is submultiplicative with respect to the
L
1
norm, making
L
1
(
G
)
a Banach algebra. The Banach algebra
L
1
(
G
)
has a multiplicative identity element if and only if G is a discrete group, namely the function that is 1 at the identity and zero elsewhere. In general, however, it has an approximate identity which is a net (or generalized sequence)
{
e
i
}
i
∈
I
indexed on a directed set
I
such that
f
∗
e
i
→
f
.
The Fourier transform takes convolution to multiplication, i.e. it is a homomorphism of abelian Banach algebras
L
1
(
G
)
→
C
0
(
G
^
)
(of norm ≤ 1):
F
(
f
∗
g
)
(
χ
)
=
F
(
f
)
(
χ
)
⋅
F
(
g
)
(
χ
)
.
In particular, to every group character on G corresponds a unique multiplicative linear functional on the group algebra defined by
f
↦
f
^
(
χ
)
.
It is an important property of the group algebra that these exhaust the set of nontrivial (that is, not identically zero) multiplicative linear functionals on the group algebra; see section 34 of the Loomis reference. This means the Fourier transform is a special case of the Gelfand transform.
As we have stated, the dual group of a locally compact abelian group is a locally compact abelian group in its own right and thus has a Haar measure, or more precisely a whole family of scalerelated Haar measures.
Theorem. Choose a Haar measure
μ
on
G
and let
ν
be the dual measure on
G
^
as defined above. If
f
:
G
→
C
is continuous with compact support then
f
^
∈
L
2
(
G
^
)
and
∫
G

f
(
x
)

2
d
μ
(
x
)
=
∫
G
^

f
^
(
χ
)

2
d
ν
(
χ
)
.
In particular, the Fourier transform is an
L
2
isometry from the complexvalued continuous functions of compact support on
G to the
L
2
functions on
G
^
(using the
L
2
norm with respect to μ for functions on
G and the
L
2
norm with respect to ν for functions on
G
^
).
Since the complexvalued continuous functions of compact support on G are
L
2
dense, there is a unique extension of the Fourier transform from that space to a unitary operator
F
:
L
μ
2
(
G
)
→
L
ν
2
(
G
^
)
.
and we have the formula
∀
f
∈
L
2
(
G
)
:
∫
G

f
(
x
)

2
d
μ
(
x
)
=
∫
G
^

f
^
(
χ
)

2
d
ν
(
χ
)
.
Note that for noncompact locally compact groups G the space
L
1
(
G
)
does not contain
L
2
(
G
)
, so the Fourier transform of general
L
2
functions on G is "not" given by any kind of integration formula (or really any explicit formula). To define the
L
2
Fourier transform one has to resort to some technical trick such as starting on a dense subspace like the continuous functions with compact support and then extending the isometry by continuity to the whole space. This unitary extension of the Fourier transform is what we mean by the Fourier transform on the space of square integrable functions.
The dual group also has an inverse Fourier transform in its own right; it can be characterized as the inverse (or adjoint, since it is unitary) of the
L
2
Fourier transform. This is the content of the
L
2
Fourier inversion formula which follows.
Theorem. The adjoint of the Fourier transform restricted to continuous functions of compact support is the inverse Fourier transform
L
ν
2
(
G
^
)
→
L
μ
2
(
G
)
where
ν
is the dual measure to
μ
.
In the case
G
=
T
the dual group
G
^
is naturally isomorphic to the group of integers
Z
and the Fourier transform specializes to the computation of coefficients of Fourier series of periodic functions.
If G is a finite group, we recover the discrete Fourier transform. Note that this case is very easy to prove directly.
One important application of Pontryagin duality is the following characterization of compact abelian topological groups:
Theorem. A locally compact
abelian group
G is compact if and only if the dual group
G
^
is discrete. Conversely,
G is discrete if and only if
G
^
is compact.
That G being compact implies
G
^
is discrete or that G being discrete implies that
G
^
is compact is an elementary consequence of the definition of the compactopen topology on
G
^
and does not need Pontryagin duality. One uses Pontryagin duality to prove the converses.
The Bohr compactification is defined for any topological group G, regardless of whether G is locally compact or abelian. One use made of Pontryagin duality between compact abelian groups and discrete abelian groups is to characterize the Bohr compactification of an arbitrary abelian locally compact topological group. The Bohr compactification B(G) of G is
H
^
, where H has the group structure
G
^
, but given the discrete topology. Since the inclusion map
ι
:
H
→
G
^
is continuous and a homomorphism, the dual morphism
G
∼
G
^
^
→
H
^
is a morphism into a compact group which is easily shown to satisfy the requisite universal property.
See also almost periodic function.
It is useful to regard the dual group functorially. In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms. The dual group construction of
G
^
is a contravariant functor LCA → LCA, represented (in the sense of representable functors) by the circle group
T
as
G
^
=
Hom
(
G
,
T
)
.
In particular, the double dual functor
G
→
G
^
^
is covariant.
Theorem. The dual group functor is an equivalence of categories from
LCA to
LCA^{op}.
Theorem. The double dual functor is naturally isomorphic to the identity functor on
LCA.
This isomorphism is analogous to the double dual of finitedimensional vector spaces (a special case, for real and complex vector spaces).
The duality interchanges the subcategories of discrete groups and compact groups. If R is a ring and G is a left Rmodule, the dual group
G
^
will become a right Rmodule; in this way we can also see that discrete left Rmodules will be Pontryagin dual to compact right Rmodules. The ring End(G) of endomorphisms in LCA is changed by duality into its opposite ring (change the multiplication to the other order). For example, if G is an infinite cyclic discrete group,
G
^
is a circle group: the former has
End
(
G
)
=
Z
so this is true also of the latter.
Such a theory cannot exist in the same form for noncommutative groups G, since in that case the appropriate dual object
G
^
of isomorphism classes of representations cannot only contain onedimensional representations, and will fail to be a group. The generalisation that has been found useful in category theory is called Tannaka–Krein duality; but this diverges from the connection with harmonic analysis, which needs to tackle the question of the Plancherel measure on
G
^
.
There are analogues of duality theory for noncommutative groups, some of which are formulated in the language of C*algebras.
When G is a Hausdorff abelian topological group, the group
G
^
with the compactopen topology is a Hausdorff abelian topological group and the natural mapping from G to its doubledual G^^ makes sense. If this mapping is an isomorphism, we say that G satisfies Pontryagin duality. This has been extended in a number directions beyond the case that G is locally compact.
S.Kaplan, in "Extensions of the Pontryagin duality" ("part I: infinite products", Duke Math. J. 15 (1948) 649–658, and "part II: direct and inverse limits", same journal, 17 (1950), 419–435) showed that arbitrary products and countable inverse limits of locally compact (Hausdorff) abelian groups satisfy Pontryagin duality. Note that an infinite product of locally compact noncompact spaces is not locally compact.
Later, in 1975, R.Venkataraman ("Extensions of Pontryagin Duality", Math. Z. 143, 105112) showed, among other facts, that every open subgroup of an abelian topological group which satisfies Pontryagin duality itself satisfies Pontryagin duality.
More recently, S. ArdanzaTrevijano and M.J. Chasco have extended the results of Kaplan mentioned above. They showed, in "The Pontryagin duality of sequential limits of topological Abelian groups", Journal of Pure and Applied Algebra 202 (2005), 11–21, that direct and inverse limits of sequences of abelian groups satisfying Pontryagin duality also satisfy Pontryagin duality if the groups are metrizable or k_{ω}spaces but not necessarily locally compact, provided some extra conditions are satisfied by the sequences.
However, there is a fundamental aspect that changes if we want to consider Pontryagin duality beyond the locally compact case. In E. MartinPeinador, A reflexible admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123 (1995), 3563–3566, it is proved that if G is a Hausdorff abelian topological group that satisfies Pontryagin duality and the natural evaluation pairing:
{
G
×
G
^
→
T
(
x
,
χ
)
↦
χ
(
x
)
is continuous, then G is locally compact. Thus any nonlocally compact example of Pontryagin duality is a group where the natural evaluation pairing of G and
G
^
is not continuous.