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Thread: Banach space

  1. #1
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    Banach space

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

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

    Hint: If $\displaystyle \sum_{n=1}^\infty x_n < \infty$ then $\displaystyle \sum_{n=k}^\infty x_n \rightarrow 0$ as k goes to infinity. (A fact that you can prove using the fact that $\displaystyle x_n \rightarrow 0$).
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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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    if X is complete and $\displaystyle \sum^{\infty}_{n=1}||X_n||<{\infty}$ then sequence $\displaystyle S_{k}=\sum^{k}_{n=1}X_{n}$ for $\displaystyle k\epsilon\aleph$ is Cauchy because for k>m
    $\displaystyle ||S_{k}-S_{m}||\leq\sum^{k}_{n=m+1}||X_{n}||\rightarrow0$ as $\displaystyle m,k\rightarrow0$
    therefore , $\displaystyle S=\sum^{\infty}_{n=1}X_{n}= lim_{k\rightarrow\infty}\sum^{k}_{n=1}X_{n}$ 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 $\displaystyle \sum^{\infty}_{n=1}||X_n||<{\infty}$ then sequence $\displaystyle S_{k}=\sum^{k}_{n=1}X_{n}$ for $\displaystyle k\epsilon\aleph$ is Cauchy because for k>m
    $\displaystyle ||S_{k}-S_{m}||\leq\sum^{k}_{n=m+1}||X_{n}||\rightarrow0$ as $\displaystyle m,k\rightarrow0$
    therefore , $\displaystyle S=\sum^{\infty}_{n=1}X_{n}= lim_{k\rightarrow\infty}\sum^{k}_{n=1}X_{n}$ exists in X.

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