Inner Product/Examples/Lebesgue 2-Space

Example of Inner Product
Let $\tuple {X, \Sigma, \mu}$ be a measure space.

Let $\map {L^2} \mu$ be the Lebesgue $2$-space of $\mu$.

Let $\innerprod \cdot \cdot: \map {L^2} \mu \times \map {L^2} \mu \to \C$ be the mapping defined by:


 * $\ds \innerprod f g = \int f \, \overline g \rd \mu$

Then $\innerprod \cdot \cdot$ is an inner product on $\map {L^2} \mu$.

Proof
First of all, by Hölder's Inequality for Integrals with $p = q = 2$, it follows that:


 * $\ds \int f \, \overline g \rd \mu$

is defined.

Now checking the axioms for an inner product in turn:

$(4)$ Positivity
Suppose that $\innerprod f f = 0$.

That is:


 * $\ds \int \cmod f^2 \rd \mu = 0$

Hence:


 * $\ds \int \cmod {f - 0}^2 \rd \mu = 0$

which is to say that $f = 0$ in $\map {L^2} \mu$ by definition of Lebesgue space.

Having verified all the axioms, we conclude $\innerprod \cdot \cdot$ is an inner product.