orthogonal to all the irreducible characters. Just as in the case of finite groups, the number of the irreducible representations up to isomorphism of a group equals the number of conjugacy classes of However, because a compact group has in general infinitely many conjugacy classes, this does not provide any useful information. The induced representation If is a closed subgroup of finite index in a compact group the definition of the induced representation for finite groups may be adopted. However, the induced representation can be defined more generally, so that the definition is valid independent of the index of the subgroup For this purpose let be a unitary representation of the closed subgroup The continuous induced representation is defined as follows: Let denote the Hilbert space of all measurable, square integrable functions with the property for all The norm is given by and the representation is given as the right-translation: The induced representation is then again a unitary representation. Since is compact, the induced representation can be decomposed into the direct sum of irreducible representations of Note that all irreducible representations belonging to the same isotype appear with a multiplicity equal to Let be a representation of then there exists a canonical isomorphism The Frobenius reciprocity transfers, together with the modified definitions of the inner product and of the bilinear form, to compact groups. The theorem now holds for square integrable functions on instead of class functions, but the subgroup must be closed. The Peter-Weyl Theorem Another important result in the representation theory of compact groups is the Peter-Weyl Theorem. It is usually presented and proven in harmonic analysis, as it represents one of its central and fundamental statements. The Peter-Weyl Theorem. Let be a compact group. For every irreducible representation of let be an orthonormal basis of We define the matrix coefficients for Then we have the following orthonormal basis of : We can reformulate this theorem to obtain a generalization of the Fourier series for functions on compact groups: The Peter-Weyl Theorem (Second version). There exists a natural –isomorphism in which is the set of all irreducible representations of up to isomorphism and is the representation space corresponding to More concretely: History The general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. Later the modular representation theory of Richard Brauer was