The *Pontrjagin dual* *G*^ of a lcag *G* is the
group of continuous homomorphisms from *G* to the unit circle
**C**_{1}. For instance, the duals of
**Z**/*n***Z**,
**Z**, **T**, **R**
are isomorphic with
**Z**/*n***Z**,
**T**, **Z**, **R**
respectively. For each *g* in *G*, the map
taking any *g^* in *G*^ to
*g^(g)* is a homomorphism from *G*^ to
**C**_{1}. If *G*^ is given the
weakest topology making all those homomorphisms continuous, it
turns out that *G*^ becomes an lcag itself, and *its*
Pontrjagin dual becomes canonically identified with *G*.
Note that on the other hand there is in general no canonical
identification between *G* and *G*^ even when these two
groups are isomorphic as lcag's (e.g. for
**Z**/*n***Z** or **R**).

Further properties of Pontrjagin duality: the dual of a finite
direct sum is naturally identified with the direct sum of the duals;
a continuous homomorphism *f* from *G* to *H*
yields a continuous homomorphism *f*^
from *H*^ to *G*^; this dual homomorphism *f*^
is injective if *f* is surjective, and vice versa; and
*f*^^=*f*. (These results can all be checked
either directly or with the assistance of the theorem
*G*^^=*G*.) In particular if *G* is
a closed subgroup of *H* then *G*^ is the quotient of
*H*^ by the *annihilator* of *G* in *H*^,
i.e. the subgroup consisting of homomorphisms taking every element
of *G* to the identity in **C**_{1}.
Conversely the dual of a quotient *H/G* is the annihiliator
of *G* in *H*^, provided *G* is a closed
subgroup of *H*.
(Check that this works for the subgroups
**Z** of **R**, and
*n***Z** of **Z**.)
The dual of a compact group is discrete, and vice versa
(hard to prove, but at least check it the examples we've seen).

Except for that last sentence, the properties of Pontrjagin duality
may remind you of duality of vector spaces. Indeed Pontrjagin duality
is a generalization of vector-space duality, at least for vector spaces
over **R** and
**Z**/*p***Z**. For instance,
if *V* is a real vector space then *V** may be
identified with *V*^ by regarding the functional *v**
as the homomorphism sending any *v* in *V* to
*e*^{iv*(v)}. For
**Z**/*p***Z** we instead
raise a fixed *p*-th root of unity to the power
*v**(*v*).

So, what does all this theory have to do with Fourier?
In each case we can express ``any'' function *f*
on *G* as a ``linear combination'' of its homomorphisms
to **C**_{1}, with the coefficients of
the homomorphisms *g^* being given by a ``Fourier transform''
*f^*(*g^*) of *f*. The ``linear combination''
may be a finite sum, infinite sum, or integral over *G^*
(actually it's always an ``integral'' for a sufficiently general
notion of integration), and except in the finite case the sense
in which this linear combination is known coincide with *f*
depends on what kind of function *f* is. In each case, though,
*f^*(*g^*) may be written as an inner product
(*f*,*g^*), and *f^^* is a function on *G*
proportional to *f*(-*g*).