# Why is $z^T x \le \|x\| \| z \|_*$ for dual norm in $R^n$?

This is probably very obvious, I was looking at this at this link.

It looks so much like a Cauchy-Schwarz though. And I would say it is very obvious from the definition if it wasn't for the condition that $\|x\| \le 1$ in the definition:

$$\| z \| := \sup_x\{ z^T x : \|x\| \le 1\}$$

How is that restriction still ok?

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In a word, scaling. –  Robert Israel Sep 24 '12 at 2:56