- #1

- 832

- 30

## Homework Statement

Hi there. I have to prove this inequality:

##||x||_2 \leq ||x||_1 \leq \sqrt{n} ||x||_2##

Where ##||x||_2## is the ##l_p## norm with p=2, so that:

##||x||_2=(|x_1|^2+|x_2|^2+...+|x_n|^2)^{\frac{1}{2}}##

And similarly ##||x||_1=|x_1|+|x_2|+...+|x_n|## is the ##l_1## vectorial norm.

so, the first part I think its easy (I suspect the second part is also easy, but I couldn't get through it).

I have that:

##||x||_2=(|x_1|^2+|x_2|^2+...+|x_n|^2)^{\frac{1}{2}}\leq |x_1|+|x_2|+...+|x_n|=||x||_1## which is directly satisfied by applying the triangle inequality. So I think thats done.

Now, for the other inequality I have to show that: ##||x||_1 \leq \sqrt{n} ||x||_2##

I couldn't find the way to show that, so I thought that perhaps someone here could help me.

Thanks in advance.