How do I prove that $$\|x\|_2^2 \leq \|x\|_1 \|x\|_\infty?$$
2026-04-04 02:18:48.1775269128
Prove the squared vector 2-norm is $\leq$ sum of 1-norm and infinity-norm
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This is the Holder inequality with $p=1$ and $q=\infty$. In addition, we can prove this inequality using the following way: $$\sum_{i=1}^n \vert x_i\vert^2=\sum_{i=1}^n \vert x_i\vert \vert x_i\vert\leq \max_{i}\vert x_i\vert\sum_{i=1}^n \vert x_i\vert=\Arrowvert x \Arrowvert_1 \Arrowvert x\Arrowvert_\infty.$$