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 \mathbb{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 \widehat 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
compact-open topology, viewing \widehat{G} as a subset of the space of all continuous functions from G to \mathbb{T}.). This topology is in general 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 \mathrm{Hom}(V, K), so too is the dual group \mathrm{Hom}(G, \mathbb{T}). More abstractly, these are both examples of
representable functors, being represented respectively by K and \mathbb{T}. A group that is isomorphic (as topological groups) to its dual group is called
self-dual. While the
reals and finite
cyclic groups are self-dual, the group and the dual group are not
naturally isomorphic, and should be thought of as two different groups.
Examples of dual groups The dual of \Z is isomorphic to the circle group \mathbb{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 \chi on \Z, \chi(n) = \chi(1)^n. Moreover, this formula defines a character for any choice of \chi(1) in \mathbb{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 \mathbb{T} is canonically isomorphic with \Z. Indeed, a character on \mathbb{T} is of the form z\mapsto z^n for n an integer. Since \mathbb{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\mapsto e^{i\theta r} for \theta 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 self-dual.) It follows that the
adeles are self-dual. ==Pontryagin duality==