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. Definition and examples. 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: The dual group . If formula_9 is a locally compact "abelian" group, a character of formula_9 is a continuous group homomorphism from formula_9 with values in the circle group formula_5. The set of all characters on formula_9 can be made into a locally compact abelian group, called the "dual group" of formula_9 and denoted formula_15. 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 formula_16 as a subset  of the space of all continuous functions from formula_9 to formula_5.). This topology is in general not metrizable. However, if the group formula_9 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 formula_20 over a field formula_21, the dual space is formula_22, so too is the dual group formula_23. More abstractly, these are both examples of representable functors, being represented respectively by formula_21 and formula_5. 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 formula_4 is isomorphic to the circle group formula_5. A character on the infinite cyclic group of integers formula_4 under addition is determined by its value at the generator 1. Thus for any character formula_29 on formula_4, formula_31. Moreover, this formula defines a character for any choice of formula_32 in formula_5. 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 formula_5 is canonically isomorphic with formula_4. Indeed, a character on formula_5 is of the form formula_37 for formula_38 an integer. Since formula_5 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 formula_40, is isomorphic to its own dual; the characters on formula_40 are of the form formula_42 for formula_43 a real number. With these dualities, the version of the Fourier transform to be  introduced next coincides with the classical Fourier transform on formula_40. Analogously, the group of formula_45-adic numbers formula_8 is isomorphic to its dual. (In fact, any finite extension of formula_8 is also self-dual.) It follows that the adeles are self-dual. Pontryagin duality. Pontryagin duality asserts that the functor induces an equivalence of categories between the opposite of the category of locally compact abelian groups (with continuous morphisms) and itself: Categorical properties. shows that the category LCA of locally compact abelian groups measures, very roughly speaking, the difference between the integers and the reals. More precisely, the algebraic K -theory spectrum of the category of locally compact abelian groups and the ones of Z and R lie in a homotopy fiber sequence