chapter  5
50 Pages

Linear Representations of Compact Topological Groups

WithL. S. Pontryagin

A linear representation of a topological group G is any homo-morphism of G into the topological group of non-singular matrices of some finite order. The question of the existence of an adequate system of linear representations for a locally compact topological group has been answered in the negative in the general case. The initial step in the construction is the definition in the group of an invariant measure or, what comes to the same thing, an invariant integral. The question of the existence of an adequate system of linear representations for a locally compact topological group has been answered in the negative in the general case. Since a topological group is, in particular, a topological space, it makes sense to speak of continuous functions defined on G. Once it was known that an invariant integral exists on an arbitrary compact group, their construction automatically generalized to the general compact case.