Minkowski inequality in Lp

Minkowski inequality: Let $(X,\Sigma,\mu)$ be a measure space and let $1\le p\le\infty$. For all $f,g\in L^p(\mu)$,

In particular, when $p=\infty$ this reads

This is the triangle inequality for the Lp norm and is what makes Lp spaces into normed linear spaces for $1\le p\le\infty$. For $p=\infty$ it is the triangle inequality for the essential supremum norm on essentially bounded functions .