Minkowski's Inequality

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $p\ge1$ be a real number. $\mathcal{L}^{p}(\mu)$ is the set of $p$-integrable functions and $\|f\|_{p}$, for $f\in\mathcal{L}^{p}(\mu)$, is the $p$-seminorm.

(i) Linear combinations. Let $f,g\in\mathcal{L}^{p}(\mu)$ and $a,b\in\mathbb{R}$. Then $af+bg\in\mathcal{L}^{p}(\mu)$.

(ii) Minkowski's inequality. Let $f,g\in\mathcal{L}^{p}(\mu)$. Then

$$\|f+g\|_{p}\le\|f\|_{p}+\|g\|_{p} .$$