Exercise 3.3

Vector and matrix $p$-norms are related by various inequalities, often involving the dimensions $m$ or $n$. For each of the following, verify the inequality and give an example of a nonzero vector or matrix (for general $m$, $n$) for which equality is achieved. In this problem $x$ is an $m$-vector and $A$ is an $m\times n$-matrix.
(a) $\parallel x{\text{∥}}_{\infty }\le \parallel x{\text{∥}}_2$.
(b) $\parallel x{\text{∥}}_2\le \sqrt{m}\parallel x{\text{∥}}_{\infty }$.
(c) $\parallel A{\text{∥}}_{\infty }\le \sqrt{n}\parallel A{\text{∥}}_2$.
(d) $\parallel A{\text{∥}}_2\le \sqrt{m}\parallel A{\text{∥}}_{\infty }$.