Understanding Cauchy Sequences in Banach Spaces

[dirk_mec1]

Homework Statement

http://img394.imageshack.us/img394/5994/67110701dt0.png

Homework Equations

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

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?

 

[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?

 

[dirk_mec1]

You have that backwards. ||fn-fm||_E<epsilon implies ||f'n-f'm||_infinity<epsilon.

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

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?

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?

 

[Dick]

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

 

[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?

 

[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.