# Show that $(Y,\| h\|_Y)$ is a Banach Space

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.

-
Schumy, Welcome to math.SE! It would be appreciated by many users here if you could elaborate a bit on your problem, perhaps indicating your partial progress, or where you got stuck. (For example, do you know how to show that it is a normed space, but are stuck on showing completeness?) It makes it easier to direct our answers in a helpful way and provides more motivation for many of us to want to help. Please feel free to add any thoughts you have about your question by clicking the "edit" link in the lower left of the post. –  Jonas Meyer Nov 28 '11 at 4:12