R^n/Equivalent norms/Example

On ${\displaystyle {}\mathbb {R} ^{n}}$, the Euclidean norm, the sum norm, and the maximum norm are equivalent. To see this, let ${\displaystyle {}x=\left(x_{1},\,\ldots ,\,x_{n}\right)\in \mathbb {R} ^{n}}$ be a vector, and assume without loss of generality that ${\displaystyle {}x_{1}}$ has the largest modulus among the components. Then we have the estimates

${\displaystyle {}\Vert {x}\Vert _{\rm {max}}=\vert {x_{1}}\vert ={\sqrt {x_{1}^{2}}}\leq {\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}\leq {\sqrt {nx_{1}^{2}}}={\sqrt {n}}\vert {x_{1}}\vert ={\sqrt {n}}\Vert {x}\Vert _{\rm {max}}\,,}$

and this yields essentially the equivalence between the Euclidean norm and the maximum norm.