Minkowski's Inequality

• categories: Real Analysis, Functional analysis, Theorem

Statement

Let $(X,\Sigma ,\mu )$ be a measure space, and let $1\le p<\infty$. For any measurable functions $f,g:X\to \mathbb{R}$ (or $\mathbb{C}$), we have:

$$\Vert f+g{\Vert }_p\le \Vert f{\Vert }_p+\Vert g{\Vert }_p,$$

For $p=\infty$, the inequality holds as well, with:

$$\Vert f+g{\Vert }_{\infty }\le \Vert f{\Vert }_{\infty }+\Vert g{\Vert }_{\infty }.$$

Intuition

Minkowski’s inequality generalizes the triangle inequality from finite-dimensional spaces to $L^p$ spaces. It says that in an $L^p$ space, the “length” of the sum $f+g$ is at most the sum of the “lengths” of $f$ and $g$.

Proof Outline for $1\le p<\infty$

1. Raise Both Sides to the $p$-th Power: Consider $\Vert f+g{\Vert }_p^p$ and expand it as: $$\Vert f+g{\Vert }_p^p={\int }_X|f(x)+g(x){|}^{p}d\mu .$$
2. Apply the $p$-Norm Triangle Inequality for Sums: Using the convexity of the function $t\mapsto t^p$ (for $p\ge 1$) and applying the inequality $|a+b{|}^p\le 2^{p-1}(|a{|}^p+|b{|}^p)$, we obtain: $$\Vert f+g{\Vert }_p^p\le {\int }_X{\left(|f(x)|+|g(x)|\right)}^{p}d\mu \le {\left(\Vert f{\Vert }_p+\Vert g{\Vert }_p\right)}^p.$$
3. Take the $p$-th Root: Taking the $p$-th root on both sides yields: $$\Vert f+g{\Vert }_p\le \Vert f{\Vert }_p+\Vert g{\Vert }_p.$$

Special Cases

• $p=1$: Minkowski’s inequality reduces to the triangle inequality for integrals: $${\int }_X|f(x)+g(x)|d\mu \le {\int }_X|f(x)|d\mu +{\int }_X|g(x)|d\mu .$$
• $p=2$: This case corresponds to the triangle inequality in $L^2$, often used in Hilbert space theory.