Question

Let re \((z)\) denote the real part of the complex number \(z\). Show that \(\langle,\rangle\) is an inner product on \(\mathbb{C}\) if \(\langle\mathbf{z}, \mathbf{w}\rangle=\operatorname{re}(z \bar{w})\)

Short answer

Yes, it is an inner product on \(\mathbb{C}\).

Step-by-step solution

Understanding the Inner Product

To show that a given function is an inner product, it must satisfy four properties: conjugate symmetry, linearity in the first argument, positivity, and definiteness.

Property 1: Conjugate Symmetry

For conjugate symmetry, we need to show that \[\langle \mathbf{w}, \mathbf{z} \rangle = \overline{\langle \mathbf{z}, \mathbf{w} \rangle}\]Given that \[\langle \mathbf{z}, \mathbf{w} \rangle = \operatorname{re}(z \bar{w})\]Since the real part of a complex number is equal to itself when conjugated, we have \[\langle \mathbf{w}, \mathbf{z} \rangle = \operatorname{re}(w \bar{z}) = \operatorname{re}(\overline{z \bar{w}}) = \operatorname{re}(z \bar{w})\]Thus, conjugate symmetry holds as \(\langle \mathbf{w}, \mathbf{z} \rangle = \langle \mathbf{z}, \mathbf{w} \rangle.\)

Property 2: Linearity in First Argument

For linearity, we must show \[\langle \alpha \mathbf{z} + \beta \mathbf{v}, \mathbf{w} \rangle = \alpha \langle \mathbf{z}, \mathbf{w} \rangle + \beta \langle \mathbf{v}, \mathbf{w} \rangle\]Calculating the left side, we have \[\langle \alpha z + \beta v, w \rangle = \operatorname{re}((\alpha z + \beta v)\overline{w})\]\[= \operatorname{re}(\alpha z \bar{w} + \beta v \bar{w}) = \operatorname{re}(\alpha z \bar{w}) + \operatorname{re}(\beta v \bar{w})\]\[= \alpha \operatorname{re}(z \bar{w}) + \beta \operatorname{re}(v \bar{w})\]\[= \alpha \langle z, w \rangle + \beta \langle v, w \rangle\]So linearity in the first argument holds.

Property 3: Positivity

For positivity, we show \(\langle \mathbf{z}, \mathbf{z} \rangle \geq 0\)This requires evaluating \[\operatorname{re}(z \bar{z}) = \operatorname{re}(|z|^2) = |z|^2 \geq 0\]Positivity holds because the modulus squared of a complex number is always non-negative.

Property 4: Definiteness

For definiteness, we need to prove that \[\langle \mathbf{z}, \mathbf{z} \rangle = 0 \Rightarrow \mathbf{z} = 0\]Since \(\langle z, z \rangle = |z|^2 = 0\), it implies \(|z| = 0\), meaning that \(z = 0\).Thus, definiteness holds.

Conclusion

The function \(\langle \mathbf{z}, \mathbf{w} \rangle = \operatorname{re}(z \bar{w})\) satisfies all four properties required for an inner product: conjugate symmetry, linearity in the first argument, positivity, and definiteness.

Key concepts

Conjugate Symmetry

Conjugate symmetry is one of the core properties required to ensure that a function acts as an inner product in a complex space. In simple terms, conjugate symmetry means that when you take the inner product of two complex numbers and then switch their places, you end up with the conjugate of the original product.
For this property, researchers typically write \[\langle \mathbf{w}, \mathbf{z} \rangle = \overline{\langle \mathbf{z}, \mathbf{w} \rangle}\]This indicates that swapping the positions of the elements within the inner product will result in a complex conjugate. In the given solution, the inner product is defined as \(\langle \mathbf{z}, \mathbf{w} \rangle = \operatorname{re}(z \bar{w})\).

• Because the real part of a complex number is equal to its conjugate, this function easily satisfies the conjugate symmetry property.
• This confirms that \(\langle \mathbf{w}, \mathbf{z} \rangle = \langle \mathbf{z}, \mathbf{w} \rangle\), meaning the inner product does not change upon swapping.

Linearity

Linearity in the first argument is another essential characteristic of an inner product. Linearity ensures a sort of predictability, allowing algebraic operations within the inner product to mirror those performed outside it.
For any scalars \(\alpha\) and \(\beta\), and complex numbers \(\mathbf{z}, \mathbf{v}, \) and \(\mathbf{w}\), linearity requires that:\[\langle \alpha \mathbf{z} + \beta \mathbf{v}, \mathbf{w} \rangle = \alpha \langle \mathbf{z}, \mathbf{w} \rangle + \beta \langle \mathbf{v}, \mathbf{w} \rangle\]

• In the solution, the left side is expanded as \(\operatorname{re}((\alpha z + \beta v)\overline{w})\).
• Further simplification ensures this equates to \(\alpha \langle \mathbf{z}, \mathbf{w} \rangle + \beta \langle \mathbf{v}, \mathbf{w} \rangle\), thereby verifying the property of linearity.

Establishing linearity allows us to distribute and scale within the function cleanly, preserving the algebraic structure when dealing with complex numbers.

Positivity

Positivity is a fundamental requirement that ensures that the inner product of a complex number with itself is always non-negative. Think of it as a way of measuring size or "length" in the complex space, which shouldn't be negative.
To reflect this, positivity demands:\[\langle \mathbf{z}, \mathbf{z} \rangle \geq 0\]The solution utilizes the expression \(\operatorname{re}(z \bar{z}) = |z|^2\).

• Since the modulus squared \(|z|^2\) is always non-negative, positivity is naturally satisfied.
• This property is vital as it quantifies the strength or magnitude of elements in the vector space.

Definiteness

Definiteness is what assures us that unless our vector is the zero vector, the inner product won't collapse to zero. In simpler terms, it's a test to ensure the only way a complex number can "vanish" in terms of magnitude is if it is itself zero.
The condition for definiteness is:\[\langle \mathbf{z}, \mathbf{z} \rangle = 0 \Rightarrow \mathbf{z} = 0\]From the step-by-step explanation:

• If \(\langle \mathbf{z}, \mathbf{z} \rangle = |z|^2 = 0\), it implies \(|z| = 0\) which logically concludes \(z = 0\).
• This property guarantees that the inner product possesses meaningful uniqueness, affirming that only the zero vector has a zero "size."