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Math Help - completion of space

  1. #1
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    completion of space

    Hi all,
    I can't solve the following , can anyone help?
    Let X = {  (x_{n})_{n=1}^{\infty} : \sum_{n=1}^{\infty} n^{-2}| x_{n}| < \infty } , and a norm on this space is defined by : || x || =  sup_{n > 0} |x_{n}| n^{-2} Show that the space (X , || .||) is not complete and find its completion. Obviously I need to find a complete and non-convergent sequence in this space , can anyone give an idea ? For the completion , I don't have an idea any hint is appreciated
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  2. #2
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    Quote Originally Posted by hermanni View Post
    Hi all,
    I can't solve the following , can anyone help?
    Let X = {  (x_{n})_{n=1}^{\infty} : \sum_{n=1}^{\infty} n^{-2}| x_{n}| < \infty } , and a norm on this space is defined by : || x || =  \sup_{n > 0} |x_{n}| n^{-2} Show that the space (X , || .||) is not complete and find its completion. Obviously I need to find a complete and non-convergent sequence in this space , can anyone give an idea ? For the completion , I don't have an idea any hint is appreciated
    For a Cauchy sequence that does not converge, take a look at (x^{(n)}), where x^{(n)} = (1,2,3,\ldots,n,0,0,0,\ldots).

    For the completion, you should expect it to be a space in which the formula for a point to belong to the space coincides with the formula for the norm. So I would expect the completion to be the space  \{(x_{n})_{n=1}^{\infty} : \sup_{n>0} n^{-2}| x_{n}| < \infty \}.

    Edit. No, I was wrong about the completion of the space. I think it should be  \{(x_{n})_{n=1}^{\infty} : \lim_{n\to\infty} n^{-2} x_{n} = 0 \}. That is complete in the given norm, and it contains X as a dense subspace.
    Last edited by Opalg; February 21st 2011 at 12:27 AM. Reason: previous guess was wrong
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