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 $\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty }$. For $1 \leqslant i \leqslant d$ we define $L _ { i } = \left( a _ { i , j } \right) _ { 1 \leqslant j \leqslant d }$ as the $i ^ { \mathrm { th } }$ row vector of $A$. Show that $$\| A \| = \max _ { 1 \leqslant i \leqslant d } \left\| L _ { i } \right\| _ { 1 } .$$
\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 $\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty }$. For $1 \leqslant i \leqslant d$ we define $L _ { i } = \left( a _ { i , j } \right) _ { 1 \leqslant j \leqslant d }$ as the $i ^ { \mathrm { th } }$ row vector of $A$. Show that
$$\| A \| = \max _ { 1 \leqslant i \leqslant d } \left\| L _ { i } \right\| _ { 1 } .$$