Hölder's inequality

Remarks

Proof idea: if a factor on the right is $0$ or $\infty$ the claim is easy. Otherwise scale $f$ and $g$ so that both factors are $1$, apply Young's inequality $ab \le \frac{a^p}{p} + \frac{b^q}{q}$ pointwise to $\abs{f(s)}$ and $\abs{g(s)}$, and integrate. When the right side is finite, equality holds if and only if there are constants $\alpha, \beta \ge 0$, not both $0$, with $\alpha \abs{f}^p = \beta \abs{g}^q$ almost everywhere.

With the counting measure on $\set{1, \dots, n}$ the statement becomes Hölder's inequality for sums. For $p = q = 2$ it is the Cauchy–Schwarz inequality for integrals. The same bound holds for $p = 1$ and $q = \infty$, with the essential supremum of $\abs{g}$ in place of the second factor.

Going deeper

For a student

In the language of Lp spaces, it says $\norm{f g}_1 \le \norm{f}_p \norm{g}_q$. So each $g \in L^q$ gives a bounded linear functional on $L^p$, which is the start of the duality between $L^p$ and $L^q$. It is also the key step in the proof of the Minkowski inequality, the triangle inequality in $L^p$.