Theorem 12.1.5 (Hölder’s Inequality, [6.2, Fol99]). Let $(X, \cm, \mu)$ be a measure space, $E, F$ be a normed vector spaces, $p, q \in [1, \infty]$. If $p, q$ are Hölder conjugates or if $p = 1$ and $q = \infty$, then for any $f \in \mathcal{L}^{p}(X; E)$ and $g \in \mathcal{L}^{q}(X; F)$,
\[\int \norm{f}_{E} \norm{g}_{F} d\mu \le \norm{f}_{L^p(X; E)}\norm{g}_{L^q(X; F)}\]
Proof. First suppose that $p = 1$ and $q = \infty$. In this case,
\[\int \norm{f}_{E} \norm{g}_{F} d\mu \le \norm{g}_{L^\infty(X; F)}\int \norm{f}_{E}d\mu = \norm{f}_{L^1(X; E)}\norm{g}_{L^\infty(X; F)}\]
Now suppose that $p, q \in (1, \infty)$ are Hölder conjugates. Assume without loss of generality that $\norm{f}_{L^p(X; E)}= \norm{g}_{L^q(X; F)}= 1$. By Young’s inequality,
\[\int \norm{f}_{E} \norm{g}_{F} d\mu \le \int \frac{\norm{f}_{E}^{p}}{p}+ \frac{\norm{g}_{F}^{q}}{q}d\mu = \frac{1}{p}\int \norm{f}_{E} d\mu + \frac{1}{q}\int \norm{g}_{F}^{q} d\mu = 1\]
$\square$