class functions. If there were some that is not the highest weight of a representation, then the corresponding would not be the character of a representation. Thus, the characters would be a proper subset of the set of 's. But then we have an impossible situation: an orthonormal basis'' (the set of characters of the irreducible representations) would be contained in a strictly larger orthonormal set (the set of 's). Thus, every must actually be the highest weight of a representation. Duality The topic of recovering a compact group from its representation theory is the subject of the Tannaka–Krein duality, now often recast in terms of Tannakian category theory. From compact to non-compact groups The influence of the compact group theory on non-compact groups was formulated by Weyl in his unitarian trick. Inside a general semisimple Lie group there is a maximal compact subgroup, and the representation theory of such groups, developed largely by Harish-Chandra, uses intensively the restriction of a representation to such a subgroup, and also the model of Weyl's character theory. See also Peter–Weyl theorem Maximal torus Root system Locally compact group p-compact group Protorus Classifying finite-dimensional representations of Lie algebras Weights in the representation theory of semisimple Lie algebras References Bibliography Topological groups Lie groups Fourier analysis