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== Inner products ==
The set of class functions of a group ''G'' with values in a field ''K'' form a ''K''-[[vector space]]. If ''G'' is finite and the [[characteristic (algebra)|characteristic]] of the field does not divide the order of ''G'', then there is an [[inner product]] defined on this space defined by <math> \langle \phi , \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \phi(g) \psi(g^{-1}) </math> where |''G''| denotes the order of ''G''. The set of [[Character theory|irreducible characters]] of ''G'' forms an [[orthogonal basis]], and if ''K'' is a [[splitting field]] for ''G'', for instance if ''K'' is [[algebraically closed]], then the irreducible characters form an [[orthonormal basis]].
In the case of a [[compact group]] and ''K'' = '''C''' the field of [[complex number]]s, the notion of [[Haar measure]] allows one to replace the finite sum above with an integral: <math> \langle \phi, \psi \rangle = \int_G \phi(t) \psi(t^{-1})\, dt. </math>
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