Can we say that; if the $l_1$-norm of an arbitrary vector $a$ is smaller that $l_1$-norm of $b$ ($||a||_1 \le ||b||_1$) then the $l_2$-norm of $a$ is smaller than $l_2$-norm of $b$ ($||a||_2 \le ||b||_2$) as well?
Thanks
Can we say that; if the $l_1$-norm of an arbitrary vector $a$ is smaller that $l_1$-norm of $b$ ($||a||_1 \le ||b||_1$) then the $l_2$-norm of $a$ is smaller than $l_2$-norm of $b$ ($||a||_2 \le ||b||_2$) as well?
Thanks
Consider $\Bbb R^2$, $a=(1,3)$ and $b=(2,2)$. They have the same $l^1$ norm, but their $l^2$-norm is different.