Definition 12.1.7 ($L^{p}$ Space). Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, and $p \in [1, \infty]$, then $\norm{\cdot}_{L^p(X; E)}$ is a seminorm on $\mathcal{L}^{p}(X; E)$. The quotient
\[L^{p}(X, \cm, \mu; E) = \mathcal{L}^{p}(X, \cm, \mu; E)/\bracs{f|f = 0\text{ a.e.}}\]
is a normed vector space, known as the $E$-valued $L^{p}$ space on $(X, \cm, \mu)$.
Proof. By Minkowski’s Inequality.$\square$