Linear Algebra and Its Applications · Problem 7.2.20

Computations with Matrices — Gilbert Strang

Mathematics · vector-norms-triangle-inequality

Problem

All vector norms must satisfy the triangle inequality. Prove that

\|x+y\|_\infty \leq \|x\|_\infty + \|y\|_\infty \qquad \text{and} \qquad \|x+y\|_1 \leq \|x\|_1 + \|y\|_1.