L-2 Space forms Hilbert Space

Theorem
Let $\struct{ X, \Sigma, \mu }$ be a measure space.

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

Let $\innerprod \cdot \cdot$ be the inner product on $\map {L^2} \mu$.

Then $\map {L^2} \mu$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space.