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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?

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    In a word, scaling.2012-09-24
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    http://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality2012-09-24

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