# Infinity norm

1. Oct 2, 2008

### dirk_mec1

1. The problem statement, all variables and given/known data

Can I take c such that

$$c = \frac{2 ||f ||_{\infty} }{ ||f||_E}$$

and in case f= 0 everywhere c=1?

2. Oct 2, 2008

### Dick

The point to c being a constant is that it's independent of f, right? Try and read the question eliminating the 'norm' words. You have a function f(x) that has a bounded derivative on [0,1] and f(0)=0. Can you get a bound for f(x) in terms of the bound on the derivative?

3. Oct 2, 2008

### dirk_mec1

Yes, you're right!

Well I know that:

$$f'(x)= \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$

But I don't see how that is going to help...

4. Oct 2, 2008

### Dick

If you are studying Banach spaces, you must know more about derivatives than just the definition. If you know f'(x) how can you find f(x)?

5. Oct 4, 2008

### dirk_mec1

$$f(x) = \int_0^x f'(t)\ \mbox{d}t$$

I carefully looked two times in my notes from my instructor and I can't find anything that relates f to f' (in context of Banach spaces). Can you give please me another hint, Dick?

Last edited: Oct 4, 2008
6. Oct 4, 2008

### morphism

Take the absolute value of both sides of this, and then work from there.

7. Oct 4, 2008

### dirk_mec1

Do you mean like this?

$$|f(x)| = |\int_0^x f'(t)\ \mbox{d}t| \leq \int_0^x |f'(t)|\ \mbox{d}t \leq \int_0^x |f'(t)|_{\infty} \ \mbox{d}t \leq \int_0^1 M \mbox{d}t = M$$

Is this correct?

8. Oct 4, 2008

### Dick

That's what I've been waiting for.