# Banach space

by dirk_mec1
Tags: banach, space
 P: 659 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 $$(f_n)_n$$ 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?
 Sci Advisor HW Helper Thanks P: 24,989 You have that backwards. ||fn-fm||_E
P: 659
 Quote by Dick You have that backwards. ||fn-fm||_E
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?