Locally compact abelian group

In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the discrete topology), or the real numbers or the circle (both with their usual topology) are locally compact abelian groups. A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff; the topological group is called abelian if the underlying group is abelian. Examples of locally compact abelian groups include: for n a positive integer, with vector addition as group operation. The positive real numbers with multiplication as operation. This group is isomorphic to 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 under addition, again with the discrete topology. The circle group, denoted for torus. This is the group of complex numbers of modulus 1. is isomorphic as a topological group to the quotient group . The field of p-adic numbers under addition, with the usual p-adic topology. If is a locally compact abelian group, a character of is a continuous group homomorphism from with values in the circle group . The set of all characters on can be made into a locally compact abelian group, called the dual group of and denoted . 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 as a subset of the space of all continuous functions from to .). This topology is in general not metrizable. However, if the group 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 over a field , the dual space is , so too is the dual group .