Riesz-Fischer Theorem

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $p \in \R$, $p \ge 1$.

The Lebesgue $p$-space $L^p \left({\mu}\right)$, endowed with the $p$-norm, is a complete metric space.

Proof
Fischer proved the result for $p = 2$, while Riesz (independently) proved it for all $p \ge 1$.