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Let $E$ a Banach Space. Let Y:=\{h:E\to\mathbb{R} \ : \ h \text{ bounded, Fréchet differentiable and Lipschitz} \} . Let \|h\|_Y:=\|h\|_{\infty}+\|h'\|_{\infty}. Show that $(Y,\|\cdot\|_Y)$ is a Banach Space.

Edit: Ah, thanks for the tip, I have proved that is a norm only :S and I need the completeness.

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    http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem2011-11-29

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