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Infinity norm

  1. Oct 2, 2008 #1
    1. The problem statement, all variables and given/known data

    [​IMG]

    Can I take c such that

    [tex]
    c = \frac{2 ||f ||_{\infty} }{ ||f||_E}
    [/tex]

    and in case f= 0 everywhere c=1?
     
  2. jcsd
  3. Oct 2, 2008 #2

    Dick

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    The point to c being a constant is that it's independent of f, right? Try and read the question eliminating the 'norm' words. You have a function f(x) that has a bounded derivative on [0,1] and f(0)=0. Can you get a bound for f(x) in terms of the bound on the derivative?
     
  4. Oct 2, 2008 #3
    Yes, you're right!

    Well I know that:

    [tex]f'(x)= \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} [/tex]

    But I don't see how that is going to help...
     
  5. Oct 2, 2008 #4

    Dick

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    If you are studying Banach spaces, you must know more about derivatives than just the definition. If you know f'(x) how can you find f(x)?
     
  6. Oct 4, 2008 #5
    [tex]f(x) = \int_0^x f'(t)\ \mbox{d}t [/tex]

    I carefully looked two times in my notes from my instructor and I can't find anything that relates f to f' (in context of Banach spaces). Can you give please me another hint, Dick?
     
    Last edited: Oct 4, 2008
  7. Oct 4, 2008 #6

    morphism

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    Take the absolute value of both sides of this, and then work from there.
     
  8. Oct 4, 2008 #7
    Do you mean like this?

    [tex]|f(x)| = |\int_0^x f'(t)\ \mbox{d}t| \leq \int_0^x |f'(t)|\ \mbox{d}t \leq \int_0^x |f'(t)|_{\infty} \ \mbox{d}t \leq \int_0^1 M \mbox{d}t = M
    [/tex]

    Is this correct?
     
  9. Oct 4, 2008 #8

    Dick

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    That's what I've been waiting for.
     
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