representations of compact groups are equivalent to unitary representations
Theorem - Let be a compact topological group. If is a finite-dimensional representation (http://planetmath.org/TopologicalGroupRepresentation) of in a normed vector space , then is equivalent (http://planetmath.org/TopologicalGroupRepresentation) to a unitary representation.
Now we claim that, for every , is a unitary operator for this new inner product. This is true since
Denote by the space endowed the inner product . As we have seen, is a unitary representation of in . Of course, and are equivalent representations, since
where is the identity mapping. Thus, is equivalent to a unitary representation.
|Title||representations of compact groups are equivalent to unitary representations|
|Date of creation||2013-03-22 18:02:28|
|Last modified on||2013-03-22 18:02:28|
|Last modified by||asteroid (17536)|