Lemma 2.7. Assuming that:

Then

(i) ${∥ f ∥}_{L^{2} (G)}^{2} = {∥ \hat{f} ∥}_{l^{2} (\hat{G})}^{2}$ (Parseval’s identity)
(ii) $⟨ f, g ⟩ = ⟨ \hat{f}, \hat{g} ⟩$ (Plancherel’s identity)