Theorem 16.1.6 (Hölder’s Inequality).label Let $(X, \cm, \mu)$ be a measure space and $p, q \in [1, \infty]$ be Hölder conjugates, then for any $f, g: \in L^{+}(X)$,
\[\int fg d\mu \le \norm{f}_{L^p(X; \real)}\norm{g}_{L^q(X; \real)}\]
Proof, [Theorem 6.2, Fol99]. First suppose that $p = 1$ and $q = \infty$. In this case,
\[\int fg d\mu \le \norm{g}_{L^\infty(X; \real)}\int fd\mu = \norm{f}_{L^1(X; \real)}\norm{g}_{L^\infty(X; \real)}\]
Now suppose that $p, q \in (1, \infty)$ are Hölder conjugates, and assume without loss of generality that $\norm{f}_{L^p(X; \real)}= \norm{g}_{L^q(X; \real)}= 1$. In which case, by Young’s inequality,
\[\int fg d\mu \le \int \frac{f^{p}}{p}+ \frac{g^{q}}{q}d\mu = \frac{1}{p}\int f d\mu + \frac{1}{q}\int g d\mu = 1\]
$\square$
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