is again a compact group when provided with the product topology. The Haar measure on the direct product is then given by the product of the Haar measures on the factor groups. For the dual representation on compact groups we require the topological dual of the vector space This is the vector space of all continuous linear functionals from the vector space into the base field. Let be a representation of a compact group in The dual representation is defined by the property Thus, we can conclude that the dual representation is given by for all The map is again a continuous group homomorphism and thus a representation. On Hilbert spaces: is irreducible if and only if is irreducible. By transferring the results of the section decompositions to compact groups, we obtain the following theorems: Theorem. Every irreducible representation of a compact group into a Hilbert space is finite-dimensional and there exists an inner product on such that is unitary. Since the Haar measure is normalized, this inner product is unique. Every representation of a compact group is isomorphic to a direct Hilbert sum of irreducible representations. Let be a unitary representation of the compact group Just as for finite groups we define for an irreducible representation the isotype or isotypic component in to be the subspace This is the sum of all invariant closed subspaces which are –isomorphic to Note that the isotypes of not equivalent irreducible representations are pairwise orthogonal. Theorem. (i) is a closed invariant subspace of (ii) is –isomorphic to the direct sum of copies of (iii) Canonical decomposition: is the direct Hilbert sum of the isotypes in which passes through all the isomorphism classes of the irreducible representations. The corresponding projection to the canonical decomposition in which is an isotype of is for compact groups given by where and is the character corresponding to the irreducible representation Projection formula For every representation of a compact group we define In general is not –linear. Let The map is defined as endomorphism on by having the property which is valid for the inner product of the Hilbert space Then is –linear, because of where we used the invariance of the Haar measure. Proposition. The map is a projection from to If the representation is finite-dimensional, it is possible to determine the direct sum of the trivial subrepresentation just as in the case of finite groups. Characters, Schur's