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Show that a vector space is not complete (therefore not a Hilbert spac

  1. Apr 7, 2013 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    Consider the space of continuous functions in [0,1] (that is C([0,1]) over the complex numbers with the following scalar product: ##\langle f , g \rangle = \int _0 ^1 \overline{f(x)}g(x)dx##.
    Show that this space is not complete and therefore is not a Hilbert space.
    Hint:Find a Cauchy sequence (with respect to the norm ##||f||=\sqrt {\langle f,f \rangle }## of functions in C([0,1])) such that the sequence converges to a non continuous function.


    2. Relevant equations
    Hmm.


    3. The attempt at a solution

    So I have in mind a sequence that reprent a truncated Fourier series that would represent a square wave if the series is never truncated. A kind of Heaviside function.
    Or a sequence of Gaussians that would reprent the Dirac delta (but it's not a function).
    In that case ##f_n=Ce^{-ax^2n^2}## would make it? I'm somehow confused when n tends to infinity to what happens at x=0.
     
  2. jcsd
  3. Apr 7, 2013 #2

    Dick

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    Try something much simpler, like ##f_n=x^n##. What does that do?
     
  4. Apr 7, 2013 #3

    fluidistic

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    Hmm since x is between 0 and 1, the sequence would converge to 0.
    Edit: Ah yeah at x=1 it's worth 1, a discontinuous function. Wow.
     
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