> [!NOTE] Minkowksi's Inequality for Euclidean space > Let $1\leq p\leq \infty$ then for all $x, y\in \mathbb{R}^n$, $\lVert x+y \rVert_{p} \leq \lVert x \rVert_{p} + \lVert y \rVert_{p} $where $\lVert \cdot \rVert_{p}$ denotes the [[p-Norm]] on Euclidean space. > [!NOTE] Minkowksi's Inequality for $\ell^p$ space > Contents ###### Proof of Minkowksi's Inequality for $\mathbb{R}^n$: Use [[Equivalence of Triangle Inequality and Convexity of Closed Unit Ball for Norms]]. ###### Proof of Minkowksi's Inequality for $\ell^p$: Follows from taking limits as $n \to \infty$ in [[Minkowksi's Inequality]].