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