lemma and the inner product Generally, representations of compact groups are investigated on Hilbert- and Banach spaces. In most cases they are not finite-dimensional. Therefore, it is not useful to refer to characters when speaking about representations of compact groups. Nevertheless, in most cases it is possible to restrict the study to the case of finite dimensions: Since irreducible representations of compact groups are finite-dimensional and unitary (see results from the first subsection), we can define irreducible characters in the same way as it was done for finite groups. As long as the constructed representations stay finite-dimensional, the characters of the newly constructed representations may be obtained in the same way as for finite groups. Schur's lemma is also valid for compact groups: Let be an irreducible unitary representation of a compact group Then every bounded operator satisfying the property for all is a scalar multiple of the identity, i.e. there exists such that Definition. The formula defines an inner product on the set of all square integrable functions of a compact group Likewise defines a bilinear form on of a compact group The bilinear form on the representation spaces is defined exactly as it was for finite groups and analogous to finite groups the following results are therefore valid: Theorem. Let and be the characters of two non-isomorphic irreducible representations and respectively. Then the following is valid i.e. has "norm" Theorem. Let be a representation of with character Suppose is an irreducible representation of with character The number of subrepresentations of equivalent to is independent of any given decomposition for and is equal to the inner product Irreducibility Criterion. Let be the character of the representation then is a positive integer. Moreover if and only if is irreducible. Therefore, using the first theorem, the characters of irreducible representations of form an orthonormal set on with respect to this inner product. Corollary. Every irreducible representation of is contained –times in the left-regular representation. Lemma. Let be a compact group. Then the following statements are equivalent: is abelian. All the irreducible representations of have degree Orthonormal Property. Let be a group. The non-isomorphic irreducible representations of form an orthonormal basis in with respect to this inner product. As we already know that the non-isomorphic irreducible representations are orthonormal, we only need to verify that they generate This may be done, by proving that there exists no non-zero square integrable function on