Now if f is an element of C c (G), the operator π π ν ν , k ( f ) defined by π π ν ν , k ( f ) = ∫ ∫ G f ( g ) π π ( g ) d g is Hilbert–Schmidt. Define a Hilbert space H by H = ⨁ ⨁ k ⩾ ⩾ 0 HS ( L 2 ( C ) ) ⊗ ⊗ L 2 ( R , c k ν ν 2 + k 2 d ν ν ) , where c k = { 1 4 π π 3 / 2 k = 0 1 ( 2 π π ) 3 / 2 k ≠ ≠ 0 and HS ( L 2 ( C ) ) denotes the Hilbert space of Hilbert–Schmidt operators on L 2 ( C ) . Then the map U defined on C c (G) by U ( f ) ( ν ν , k ) = π π ν ν , k ( f ) extends to a unitary of L 2 ( G ) onto H.