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Math Help - complete matric space

  1. #1
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    complete matric space

    Let X be the set of all polynomial functions of degree 4 or less on [0,1]. If f=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4 and g=b_0+b_1x+b_2x^2+b_3x^3+b_4x^4, we define d_2(f(x),g(x))=\sum_{n=0}^4 \mid a_n-b_n \mid.
    Further, we define \rho_2(f(x),g(x))=\frac{d_2(f(x),g(x))}{1+d_2(f(x)  ,g(x))}.

    Will (X,\rho_2) be a complete metric space?
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  2. #2
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    Since X is finite dimensional and d_2 is a norm it follows that (X,d_2) is a Banach space.

    If (x_n)_n is a Cauchy sequence in (X,\rho_2), then for \varepsilon>0 there exist n_0 such that \rho_2(x_n,x_m)<\varepsilon for n,m>n_0. A direct calculation shows that this implies

    d_2(x_n,x_m)<\frac{\varepsilon}{1-\varepsilon}.

    Taking \varepsilon small enough (the quotient in the last inequality goes to 0 as \varepsilon goes to 0), this shows that (x_n)_n is a Cauchy sequence in (X,d_2). Hence there
    x\in X such that x_n tends to x in (X,d_2). But this implies that (x_n)_n tends to x in (X,\rho_2), since
    \rho_2\leq d_2.

    Thus the answer is yes if you show that \rho_2 is actually a metric.
    Last edited by Enrique2; August 13th 2009 at 01:11 AM. Reason: Latex mistakes
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  3. #3
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    cauchy sequence

    how do you show that a cauhy sequence is convergent in (X,d_2) or (X, \rho_2)?
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  4. #4
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    Quote Originally Posted by Kat-M View Post
    how do you show that a cauhy sequence is convergent in (X,d_2) or (X, \rho_2)?
    Showing that d_2 is a norm, on X, since X is finite dimensional, you obtain that (X,d_2) is complete. You can consult any book related to topological vector spaces, in Robertson and Robertson's book "Topological vector spaces" for instance. Perhaps you are not allowed to use this fact?

    The completenes of \rho_2 follow from the given argument above, Cauchy in \rho_2 implies Cauchy in d_2, then convergence in d_2 and \rho_2\leq d_2.

    I believe that in the same book I have mentioned you can find the clues to show that \rho_2 defines a metric.
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