Measure, Integration & Real Analysis, 2021a

158 Chapter 6 Banach Spaces
We now define the complex conjugate of a complex number.
6.23 Definition complex conjugate; z
Suppose z ∈ C. The complex conjugate of z ∈ C, denoted z (pronounced z-bar),
is defined by
z = Re z − (Im z)i.
For example, if z = 5 + 7i then z = 5 − 7i. Note that a complex number z is a
real number if and only if z = z.
The next result gives basic properties of the complex conjugate.
6.24 properties of complex conjugates
Suppose w, z ∈ C. Then
• product of z and z
z z = |z| 2 ;
• sum and difference of z and z
z + z = 2Rez and z − z = 2(Im z)i;
• additivity and multiplicativity of complex conjugate
w + z = w + z and wz = w z;
• complex conjugate of complex conjugate
z = z;
• absolute value of complex conjugate
|z| = |z|;
• integral of complex conjugate of a function
∫ ∫
f dμ = f dμ for every measure μ and every f ∈L 1 (μ).
Proof
The first item holds because
zz =(Re z + i Im z)(Re z − i Im z) =(Re z) 2 +(Im z) 2 = |z| 2 .
To prove the last item, suppose μ is a measure and f ∈L 1 (μ). Then
∫ ∫
∫
∫
f dμ = (Re f − i Im f ) dμ = Re f dμ − i Im f dμ
∫
=
∫
=
∫
Re f dμ + i
f dμ,
Im f dμ
as desired.
The straightforward proofs of the remaining items are left to the reader.
Measure, Integration & Real Analysis, by Sheldon Axler