1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Banach space

  1. Sep 27, 2008 #1
    1. The problem statement, all variables and given/known data
    [​IMG]


    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?
     
  2. jcsd
  3. Sep 27, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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?
     
  4. Sep 28, 2008 #3
    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?
     
  5. Sep 28, 2008 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, that's it. Show f exists and has bounded derivative.
     
  6. Sep 29, 2008 #5
    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?
     
  7. Sep 29, 2008 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Banach space
  1. Banach space problem (Replies: 1)

Loading...