Hölder's Inequality

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space. For a real $r>1$, $\mathcal{L}^{r}(\mu)$ is the set of $r$-integrable functions and $\|f\|_{r}$, for $f\in\mathcal{L}^{r}(\mu)$, is the $r$-seminorm.

(i) Hölder's inequality. Let $p,q>1$ be real numbers with $1/p+1/q=1$, and let $f\in\mathcal{L}^{p}(\mu)$ and $g\in\mathcal{L}^{q}(\mu)$. Then $fg\in\mathcal{L}^{1}(\mu)$ and

$$\int|fg|\,d\mu\le\|f\|_{p}\,\|g\|_{q},$$

the integral of $|fg|$ as an element of $\mathcal{M}^{+}$.

(ii) The Cauchy-Schwarz inequality. Let $f,g\in\mathcal{L}^{2}(\mu)$. Then $fg\in\mathcal{L}^{1}(\mu)$ and

$$\Bigl|\int fg\,d\mu\Bigr|\le\|f\|_{2}\,\|g\|_{2} .$$