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Michal Grňo (talk | contribs) m →Inner products: added link to Splitting field |
Hrodelbert (talk | contribs) →Inner products: the inner products as they were defined were not positive definite on the space of class functions, consider for instance the class function \psi(g) = i. On characters both definitions (old and new) coincide, but only the new one is an inner product. |
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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) \overline{\psi(g
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) \overline{\psi(t
When ''K'' is the real numbers or the complex numbers, the inner product is a [[degenerate form|non-degenerate]] [[Hermitian form|Hermitian]] [[bilinear form]].
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