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Sequence Convergence

  1. Feb 26, 2010 #1
    If a sequence [tex]\{f_n\}[/tex] is convergent in [tex]\left(C[0,1],||\cdot||_{\infty}\right)[/tex] then it is also convergent in [tex]\left(C[0,1],||\cdot||_1\right)[/tex].

    I think I understand why this is true. (In my own words) The relationship between the supremum norm and the usual norm (really any p-norm) is that the supremum norm is the greatest value in all p-norms. So, for all sequences, if the sequence is convergent in the supremum norm it's convergent in all norms on the same space. Is this true?

    Also, for a sequence to converge it means
    [tex]\exists \, f \,\ni \,\forall \,\epsilon>0 \,\exists\, N\ni\, \forall\, x\in C[0,1][/tex]
    [tex]||f_n-f||_{\infty}<\epsilon \quad \forall n>N[/tex]

    This is a given, but how could I use that to prove the implied part? This is for my own edification.

    Also, I can think of a counter example to show the other direction is not true.
    Such as, [tex]f_n(t)=t^n \quad \text{then} \quad ||f_n||_1\rightarrow 0[/tex]
    but, [tex]||f_n||_{\infty}\rightarrow 1[/tex]
  2. jcsd
  3. Feb 26, 2010 #2


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    Re: Convergence

    The first statement is not true in general, only for measure spaces where the total measure is finite.

    Counterexample on the real line:

    fn(x)=1 -n<x<n, =0 |x|>n+1, and connect up to be continuous in between.
    sup |fn(x)|=1, while all Lp norms become infinite.
  4. Feb 26, 2010 #3
    Re: Convergence

    Right, thank you. That is a good point. I haven't yet thought about this proclamation in all spaces. My statement was about the behavior was concerning this normed metric space.
  5. Feb 27, 2010 #4
    Re: Convergence

    Since (C[0,1],[tex]||\cdot||_{\infty}[/tex]) is complete and [tex]\{f_n\}[/tex] is convergent,
    we know every sequence in (C[0,1],[tex]||\cdot||_{\infty}[/tex]) is Cauchy convergent and converges uniformly [tex]\Rightarrow \quad ||f_n-f||_{\infty}\rightarrow \, 0[/tex].
    Because of this we also know:
    [tex]||f_n-f||_1\rightarrow \, 0[/tex]

    Is my reasoning correct?
  6. Feb 28, 2010 #5


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    Re: Convergence

    Yes, You can use the same idea for all Lp norms.
  7. Feb 28, 2010 #6
    Re: Convergence

    Thanks, I appreciate your input.
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