Exercise10.5
Show that the following inequalities hold for all vectors \(x \in \Cm\text{.}\) If possible, give a nontrivial example for which equality holds.
- \(\|x\|_\infty \leq \| x\|_2\)
- \(\|x\|_2 \leq \sqrt{m}\|x\|_\infty\)
- \(\|x\|_2 \leq \sqrt{\|x\|_1 \|x\|_\infty}\)
- \(\|x\|_\infty \leq \|x\|_2 \leq \|x\|_1\)
Which, if any, of these inequalities extend to the matrix norms induced by these vector norms?