 # How do I find a constant B>0, so that norm(x) <= B * norm(x)_oo works for all x in RR^n? Jaylyn Gibson 2022-08-07 Answered
How do I find a constant B>0, so that $‖x‖$ $\le$ $B\ast {‖x‖}_{\mathrm{\infty }}$ works for all $x\in {\mathbb{R}}^{n}$?
Not sure but I think I have to look at $x$ = $\sum _{i=1}^{n}{x}_{i}{e}_{i}$ with the development of the x to the canonical unit vectors ${e}_{i}$. But how do I do that?
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Write $x={x}_{1}{e}_{1}+\cdots +{x}_{n}{e}_{n}$. If $‖\cdot ‖$ is an arbitrary norm on ${\mathbb{R}}^{n}$ you have
$‖x‖=‖{x}_{1}{e}_{1}+\dots {x}_{n}{e}_{n}‖\le \sum _{i=1}^{n}‖{x}_{i}{e}_{i}‖=\sum _{i=1}^{n}|{x}_{i}|‖{e}_{i}‖.$
Each ${x}_{i}$ satisfies $|{x}_{i}|\le ‖x{‖}_{\mathrm{\infty }}$ so that
$‖x‖=‖{x}_{1}{e}_{1}+\dots {x}_{n}{e}_{n}‖\le \sum _{i=1}^{n}‖x{‖}_{\mathrm{\infty }}‖{e}_{i}‖=\left(\sum _{i=1}^{n}‖{e}_{i}‖\right)‖x{‖}_{\mathrm{\infty }}.$
Take
$B=\sum _{i=1}^{n}‖{e}_{i}‖.$