# Comparing norms on $\mathbb{R}^n$

We know that $\mathbb{R}^n$ is normed linear space with respect to the norms defined as follows

$\Vert x\Vert_{1} = \sum_{i =1}^n |x_i|$

$\Vert x\Vert_{2} = (\sum_{i =1}^n |x_i|^2)^{1/2 }$

$\Vert x\Vert_{\infty} = \max _{1\leq i\leq n}\{|x_i|\}$

I have studied that in general for any $x\in \mathbb{R}^n$

$\Vert x\Vert_{1}\geq \Vert x\Vert_{2}\geq \Vert x\Vert_{\infty}$ ....$(1)$

I haven't been able to prove result $(1)$. Although, I have tried few numerical examples to verify this.

Please help me and thanks for all.

## 1 Answer

See this answer to compare norms with finite parameter. Inequalities $\Vert x\Vert_p\geq\Vert x\Vert_\infty$ for finite $p$ are obvious.