\textbf{Problem 2, Part 1: Adapted norms}

We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$.

For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$ and $\| x \| _ { 1 } = \sum _ { i = 1 } ^ { d } \left| x _ { i } \right|$.

Given a matrix $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ we define
$$\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty } .$$

a. Show that this defines a norm on $\mathrm { M } _ { d } ( \mathbb { C } )$ and that there exists $x _ { 0 } \in \mathbb { C } ^ { d }$ such that $\left\| x _ { 0 } \right\| _ { \infty } = 1$ and $\left\| A x _ { 0 } \right\| _ { \infty } = \| A \|$.\\
b. Show that for all $( A , B ) \in \mathrm { M } _ { d } ( \mathbb { C } )$ we have $\| A B \| \leqslant \| A \| \cdot \| B \|$.