# Homework Help: L1 and l2 norm inequality.

1. Oct 27, 2009

### roho

1. The problem statement, all variables and given/known data
$$\|x\|_2\le\|x\|_1\le\sqrt{n}\|x\|_2$$
where |x|1 is the l1 norm and |x|2 is the l2 norm

2. Relevant equations
See above

3. The attempt at a solution
I have $$\|\mathbf{x}\|_1 := \sum_{i=1}^{n} |x_i|$$
and $$\|x\|_2 = \left(\sum_{i\in\mathbb N}|x_i|^2\right)^{\frac12}$$
I have tried to expand out the x 2 norm but i cant seem to figure out how to prove the inequality. Any suggestions?

Last edited: Oct 27, 2009
2. Oct 27, 2009

### lanedance

for the first part of the inequality, you could try squaring both sides

3. Oct 27, 2009

### roho

Yea that works for the first part. Thanks for the reply.

Any idea on the second part (square root of n)?

I am thinking it may have to do with the projection vector (such as (1,1,1,1,1,1)) in a scalar product or something like.

4. Oct 27, 2009

### lanedance

your idea should work with for the 2nd one with the use of Cauchy Schwarz