The [[character (group theory)|character]] of a [[linear representation]] of ''G'' over a [[field (mathematics)|field]] ''K'' is always a class function with values in ''K''. The class functions form the [[centerCenter (algebraring theory)|center]] of the [[group ring]] ''K''[''G'']. Here a class function ''f'' is identified with the element <math> \sum_{g \in G} f(g) g</math>.
== Inner products ==
The set of class functions of a group ''{{mvar|G''}} with values in a field ''{{mvar|K''}} form a ''{{mvar|K''}}-[[vector space]]. If ''{{mvar|G''}} is finite and the [[characteristic (algebra)|characteristic]] of the field does not divide the order of ''{{mvar|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) \overline{\psi(g^{-1})},</math> where {{math|{{!}}''G''|{{!}}}} denotes the order of ''{{mvar|G''}} and the overbar denotes conjugation in the field {{mvar|K}}. The set of [[Character theory|irreducible characterscharacter]]s of ''{{mvar|G''}} forms an [[orthogonal basis]],. andFurther, if ''{{mvar|K''}} is a [[splitting field]] for ''{{mvar|G'', }}{{--}}for instance, if ''{{mvar|K''}} is [[algebraically closed]], then the irreducible characters form an [[orthonormal basis]].
InWhen the{{mvar|G}} case ofis a [[compact group]] and {{math|''K'' {{=}} '''C'''}} is the field of [[complex number]]s, the notion of [[Haar measure]] allowscan onebe applied to replace the finite sum above with an integral: <math>\langle \phi, \psi \rangle = \int_G \phi(t) \overline{\psi(t^{-1})}\, dt.</math>
When ''{{mvar|K''}} is the real numbers or the complex numbers, the inner product is a [[degenerate form|non-degenerate]] [[Hermitian form|Hermitian]] [[bilinear form]].
