Suppose I have two vectors $z, x$ of equal size, and $|x| \leq 1$
I know the following is true:
$$\sum_iz_ix_i \leq |\sum_i z_ix_i| \leq \sum_i|z_ix_i|$$
is it legal to write the following:
$\sup_x \{ \sum_iz_ix_i\} = \sup_x \{|\sum_i z_ix_i|\} = \sup_x \{\sum_i|z_ix_i|\}$?
Why or why not.
Hint: If $z = 0$, all three suprema are $0$; otherwise, what unit vector $x$ (in terms of $z$) maximizes $\langle x, z\rangle = \sum_{i} x_{i} z_{i}$, and what is the maximum value?