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Thread: complete matric space

  1. #1
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    complete matric space

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

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

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

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

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

    Thus the answer is yes if you show that $\displaystyle \rho_2$ is actually a metric.
    Last edited by Enrique2; Aug 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 $\displaystyle (X,d_2)$ or $\displaystyle (X, \rho_2)$?
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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 $\displaystyle (X,d_2)$ or $\displaystyle (X, \rho_2)$?
    Showing that $\displaystyle d_2$ is a norm, on $\displaystyle X$, since X is finite dimensional, you obtain that $\displaystyle (X,d_2)$ 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 $\displaystyle \rho_2$ follow from the given argument above, Cauchy in $\displaystyle \rho_2$ implies Cauchy in $\displaystyle d_2$, then convergence in $\displaystyle d_2$ and $\displaystyle \rho_2\leq d_2$.

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