# Banach space

1. Sep 27, 2008

### dirk_mec1

1. The problem statement, all variables and given/known data
http://img394.imageshack.us/img394/5994/67110701dt0.png [Broken]

2. Relevant equations

A banach space is a complete normed space which means that every Cauchy sequence converges.

3. The attempt at a solution
I'm stuck at exercise (c).

Suppose $$(f_n)_n$$ is a Cauchy sequence in E. Then

$$|f_n-f_m| < \epsilon\ \forall\ n,m \leq N$$

so

$$|f'_n - f'_m| \leq |f'_n - f'_m|_{\infty} < \epsilon$$

Am I going in the right direction?

Last edited by a moderator: May 3, 2017
2. Sep 27, 2008

### Dick

You have that backwards. ||fn-fm||_E<epsilon implies ||f'n-f'm||_infinity<epsilon. Can you use the fact the difference in derivatives of fn and fm is small to prove the difference between fn and fm is small? Hence that fn(x) is a cauchy sequence for each x?

3. Sep 28, 2008

### dirk_mec1

Really? I don't see why this is so.

But what good will that do?

So here is the interpretation of the assignment in my eyes:

Given a Cauchy sequence $$(f_n)_n \in\ E$$ prove that $$||f_n-f||_E \rightarrow 0$$ and that f is in E.

So we have:

$$||f'_n -f'_m||_{\infty} < \epsilon\ \forall m,n \geq N$$

and we want: $$||f'_n -f'||_{\infty} \rightarrow 0\ \forall n \geq N$$

Is this correct?

4. Sep 28, 2008

### Dick

Yes, that's it. Show f exists and has bounded derivative.

5. Sep 29, 2008

### dirk_mec1

I'm sorry Dick, I have have been thinking about this but I can't seem to get f in E and converging.

Your posts imply that I should prove that f_n is a Cauchy sequence but what do I get from that? You also mention to use the deratives to prove that f_n is Cauchy: do you mean that I should use the definition of the derative?

6. Sep 29, 2008

### Dick

You know there is a limiting function f', since f'_n(x) is a cauchy sequence in R for every x. So that sequence has a limit, define f'(x) to be that limit. f' is also continuous since it's a uniform limit of continuous functions. Once you have f' just define f to be the integral from 0 to x of f'(t)dt.