Banach space


by dirk_mec1
Tags: banach, space
dirk_mec1
dirk_mec1 is offline
#1
Sep27-08, 10:30 AM
P: 664
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?
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
Dick
Dick is online now
#2
Sep27-08, 12:34 PM
Sci Advisor
HW Helper
Thanks
P: 25,166
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
dirk_mec1 is offline
#3
Sep28-08, 01:23 PM
P: 664
Quote Quote by Dick View Post
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 [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
Dick is online now
#4
Sep28-08, 02:18 PM
Sci Advisor
HW Helper
Thanks
P: 25,166

Banach space


Yes, that's it. Show f exists and has bounded derivative.
dirk_mec1
dirk_mec1 is offline
#5
Sep29-08, 03:22 PM
P: 664
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
Dick is online now
#6
Sep29-08, 03:42 PM
Sci Advisor
HW Helper
Thanks
P: 25,166
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.


Register to reply

Related Discussions
Banach space C(X) Differential Geometry 6
Is it possible for a Banach Space to be isomorphic to its double dual Calculus 1
Banach space vs. closed space General Math 6
Banach Spaces Linear & Abstract Algebra 4
differentiation on banach spaces Calculus 16