section ”Properties”, also apply to representations of compact groups. But there are some modifications needed: To define a subrepresentation we now need a closed subspace. This was not necessary for finite-dimensional representation spaces, because in this case every subspace is already closed. Furthermore, two representations of a compact group are called equivalent, if there exists a bijective, continuous, linear operator between the representation spaces whose inverse is also continuous and which satisfies for all If is unitary, the two representations are called unitary equivalent. To obtain a –invariant inner product from a not –invariant, we now have to use the integral over instead of the sum. If is an inner product on a Hilbert space which is not invariant with respect to the representation of then is a –invariant inner product on due to the properties of the Haar measure Thus, we can assume every representation on a Hilbert space to be unitary. Let be a compact group and let Let be the Hilbert space of the square integrable functions on We define the operator on this space by where The map is a unitary representation of It is called left-regular representation. The right-regular representation is defined similarly. As the Haar measure of is also right-translation-invariant, the operator on is given by The right-regular representation is then the unitary representation given by The two representations and are dual to each other. If is infinite, these representations have no finite degree. The left- and right-regular representation as defined at the beginning are isomorphic to the left- and right-regular representation as defined above, if the group is finite. This is due to the fact that in this case Constructions and decompositions The different ways of constructing new representations from given ones can be used for compact groups as well, except for the dual representation with which we will deal later. The direct sum and the tensor product with a finite number of summands/factors are defined in exactly the same way as for finite groups. This is also the case for the symmetric and alternating square. However, we need a Haar measure on the direct product of compact groups in order to extend the theorem saying that the irreducible representations of the product of two groups are (up to isomorphism) exactly the tensor product of the irreducible representations of the factor groups. First, we note that the direct product of two compact groups