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Math Help - Banach space

  1. #1
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    Banach space

    How would I show that the vector space \ell^1 with the 2 norm ||\cdot||_2  is not a Banach space?

    \ell ^1= ( (x_i)^\infty_{i=1} : \Sigma^\infty_{i=1}|x_i|< \infty)

    I would perhaps have to show that it doesn't converge in the appropriate norm..? I am not sure how..!
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  2. #2
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    Quote Originally Posted by bigdoggy View Post
    How would I show that the vector space \ell^1 with the 2 norm ||\cdot||_2  is not a Banach space?

    \ell ^1= ( (x_i)^\infty_{i=1} : \Sigma^\infty_{i=1}|x_i|< \infty)

    I would perhaps have to show that it doesn't converge in the appropriate norm..? I am not sure how..!
    You need to show that the space is not complete in that norm. To see that, start by taking a sequence that is in \ell^2 but not in \ell^1 . For example, you could take the element (x_i) with x_i = 1/i. Then consider the sequence of elements x^{(n)} in \ell^1 defined by \textstyle x^{(n)}(i) = \begin{cases}1/i&\text{if }i\leqslant n,\\ 0&\text{if }i>n.\end{cases}

    Then (x^{(n)}) is Cauchy for the 2-norm, but it does not have a 2-norm limit in the space \ell^1 (because it is trying to converge to an element that isn't in \ell^1 ).
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