Equivalent Norms: Proving No M > 0 for f

[dirk_mec1]

Homework Statement

http://img394.imageshack.us/img394/2907/54050356xf9.png

The Attempt at a Solution

I'm stuck at exercise (e).

What I have to proof is that there is no M>0 such that:

||f'||_{\infty} \leq M \cdot ||f||_{\infty}

But I'm having a hard time showing that for there is little information on the sup of f. One way of doing this is to show that is 'M' is not constant (at least that's what I think) but because I only know that f is in C1 and f(0)=0 I don't see a way of proving this.

Can anyone give me a hint?

 

[morphism]

Think about polynomials.

To be more specific, try to find a sequence {f_n} of polynomials in E such that \|f_n\|_\infty = 1 for all n, while \|f_n'\|_\infty \to \infty.

 

[HallsofIvy]

In order to show that a general statement is NOT true you only need a counterexample. As morphism suggested look for one among simple function, like polynomials.