The modulus squared of any complex number z can be expressed as | z | 2 = ( Re ( z ) ) 2 + ( Im ( z ) ) 2 ≥ ≥ ( Im ( z ) ) 2 = ( z − − z ∗ ∗ 2 i ) 2 . we let z = ⟨ ⟨ f | g ⟩ ⟩ and z ∗ ∗ = ⟨ ⟨ g ∣ ∣ f ⟩ ⟩ and substitute these into the equation above to get | ⟨ ⟨ f ∣ ∣ g ⟩ ⟩ | 2 ≥ ≥ ( ⟨ ⟨ f ∣ ∣ g ⟩ ⟩ − − ⟨ ⟨ g ∣ ∣ f ⟩ ⟩ 2 i ) 2 .