Prove some Equivalences Norm

Question

Suppose $X=R^2$ and $x=(x_1, x_2)$.

I see the following are equal EDIT: ( equivalence). why? i couldent find any proof to satisfy me. any hint or idea or proof highly appreciated.

$||x||_1= |x_1| + |x_2|$

$ ||x_2||_2= (|x_1|^2+|x_2|^2)^{1/2}$

$ ||x||_{max}= max${$|x_1|, |x_2|$}

Answer 1

Using $2ab\le a^2+ b^2$ and monotonicity of $x\mapsto \sqrt x$ we obtain the first $$ \|x\|_1 = \sqrt{ (|x_1|+|x_2|)^2} = \sqrt{ |x_1|^2+2|x_1|\cdot|x_2|+ |x_2|^2} \le \sqrt{ 2|x_1|^2+2|x_2|^2} = \sqrt2\|x\|_2. $$ Again by monotonicity $$ \|x\|_2 = \sqrt{ |x_1|^2+|x_2|^2} \le \sqrt{ \|x\|_{max}^2+\|x\|_{max}^2} = \sqrt2\|x\|_{max}. $$ Since $|x_i|\le |x_1|+|x_2|$ for all $i$, $$ \|x\|_{max} \le \|x\|_1. $$