# $\ell^0$ and $\ell^{\infty}$ norms

Let $x \in S^{n-1}$ and such that its coordinates $|x_1|\geq \cdots \geq |x_n|$.

Under which condition on $\|x\|_0$ the following inequality is true that $$\|x\|_{\infty}\leq \frac{1}{\sqrt {\|x\|_0}}\quad ?$$

-
Sorry but what is $\|x\|_0$? – 1015 Feb 10 '13 at 19:57
$\|x\|_0$ is a cardinality of a support of $x$. – Alex Feb 10 '13 at 19:59
Oh, I see. Thanks. – 1015 Feb 10 '13 at 19:59
Surely you can find a vector in $S^1$ with two non-zero entries the first of which is as close to $1$ as you wish. – Martin Feb 10 '13 at 20:02
@ Martin: Sorry I stated my question incorrectely. I ment is it always tru? or How to prove it? – Alex Feb 10 '13 at 20:03