In Matrix Analysis by Horn and Johnson (page 320 in 2nd Edition), the definition of the k-norm is as follow: $$\large \|x\|_{[k]} = |x_1| + \cdots\cdots + |x_k|,\text{ where } |x_1|\geq\cdots\geq|x_k|$$
And a property is as follow: $$\large \|\cdot\|_\infty = \|\cdot\|_{[1]}\leq\|\cdot\|_{[2]}\leq\cdots\leq\|\cdot\|_{[n]} = \|\cdot\|_1$$
Although I can verify this property , I am very confused.
$\|\cdot\|_\infty $ = max{ abs of the all elements} and $\|\cdot\|_1$ = {sum of all the elements}
How could they be equivalent? (Though they are equivalent through the proof) I think there is a gap. Any help is appreciable. Thanks!
P.S. You can also explain in matrix language.