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Math Help - Banach space

  1. #1
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    Banach space

    prove that, in a Banach space if  Sum||Xn|| converges then SumXn converges
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  2. #2
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    Banach space

    Can you help me please?
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  3. #3
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    prove that , in a Banach space if \sum^{\infty}_{n=1}||Xn|| converges then \sum^{\infty}_{n=1}Xn converges.
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    Quote Originally Posted by cizzzi View Post
    prove that , in a Banach space if \sum^{\infty}_{n=1}||Xn|| converges then \sum^{\infty}_{n=1}Xn converges.
    Show that the sum is Cauchy, i.e. consider S_k:=\sum_{n=1}^k X_n, now consider ||S_k-S_l||.

    Hint: If \sum_{n=1}^\infty x_n < \infty then \sum_{n=k}^\infty x_n \rightarrow 0 as k goes to infinity. (A fact that you can prove using the fact that  x_n \rightarrow 0).
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  5. #5
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    thank you I try to solve but I do not know very well functional analysis
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    Quote Originally Posted by cizzzi View Post
    thank you I try to solve but I do not know very well functional analysis
    Why don't you post what you have done so far (even if it is wrong)?
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  7. #7
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    if X is complete and \sum^{\infty}_{n=1}||X_n||<{\infty} then sequence S_{k}=\sum^{k}_{n=1}X_{n} for k\epsilon\aleph is Cauchy because for k>m
    ||S_{k}-S_{m}||\leq\sum^{k}_{n=m+1}||X_{n}||\rightarrow0 as m,k\rightarrow0
    therefore , S=\sum^{\infty}_{n=1}X_{n}= lim_{k\rightarrow\infty}\sum^{k}_{n=1}X_{n} exists in X.

    is it true?
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  8. #8
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    Quote Originally Posted by cizzzi View Post
    if X is complete and \sum^{\infty}_{n=1}||X_n||<{\infty} then sequence S_{k}=\sum^{k}_{n=1}X_{n} for k\epsilon\aleph is Cauchy because for k>m
    ||S_{k}-S_{m}||\leq\sum^{k}_{n=m+1}||X_{n}||\rightarrow0 as m,k\rightarrow0
    therefore , S=\sum^{\infty}_{n=1}X_{n}= lim_{k\rightarrow\infty}\sum^{k}_{n=1}X_{n} exists in X.

    is it true?
    The sum converges to zero as k and m tend to infinity (not zero).
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