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Proof of Minkowski's Inequality in Lebesgue Spaces 📂Lebesgue Spaces

Proof of Minkowski's Inequality in Lebesgue Spaces

Theorem[^1]

Let’s call ΩRn\Omega \subset \mathbb{R}^{n} an open set. If 1p<1 \le p < \infty and u,vLp(Ω)u, v \in L^{p}(\Omega), then

u+vpup+vp \left\| u + v \right\|_{p} \le \left\| u \right\|_{p}+\left\| v \right\|_{p}

This is called the Minkowski inequality.