Let be a Finite Abelian Group. In the space define the inner product of as: and the norm: Also for This space has an Orthonormal Basis given by the Characters. Let be the Pontryagin Dual of . In the space we define the inner product of as: and Let . The Fourier Transform of is a function given by: Lemma Parseval’s Identity: Proof Lemma Define Then Proof Lemma Proof