Minkowski inequality

Remarks

Proof idea: the case $p = 1$ follows from $\abs{f + g} \le \abs{f} + \abs{g}$. For $p > 1$, first $\abs{f + g}^p \le 2^{p-1}(\abs{f}^p + \abs{g}^p)$ shows that the left side is finite. Then write $\abs{f + g}^p \le \abs{f} \abs{f + g}^{p-1} + \abs{g} \abs{f + g}^{p-1}$ and apply Hölder's inequality to each term with $q = p / (p - 1)$.

With the counting measure on $\set{1, \dots, n}$ it shows that the p-norm on R^n satisfies the triangle inequality. In general it is the triangle inequality for the norm of the Lp space $L^p(\mu)$.

The hypothesis $p \ge 1$ is needed. For $p = 1/2$ on $\R^2$ with the counting measure, take $f = (1, 0)$ and $g = (0, 1)$: the left side is $(1 + 1)^2 = 4$ and the right side is $1 + 1 = 2$.

Going deeper

For a student

This inequality is what makes $\norm{f}_p = (\int \abs{f}^p \, d\mu)^{1/p}$ a norm for $p \ge 1$, so that $L^p$ is a normed space. Its proof is a short application of Hölder's inequality. For $0 < p < 1$ the inequality fails, and the quantity $\norm{f}_p$ is then not a norm.