Banach space

dirk_mec1

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



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 [tex] (f_n)_n [/tex] is a Cauchy sequence in E. Then

[tex] |f_n-f_m| < \epsilon\ \forall\ n,m \leq N [/tex]

so

[tex] |f'_n - f'_m| \leq |f'_n - f'_m|_{\infty} < \epsilon [/tex]


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

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 [tex](f_n)_n \in\ E[/tex] prove that [tex]||f_n-f||_E \rightarrow 0 [/tex] and that f is in E.

So we have:

[tex] ||f'_n -f'_m||_{\infty} < \epsilon\ \forall m,n \geq N [/tex]

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

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.