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

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- Jun 1st 2010, 06:42 AMcizzziBanach space
prove that, in a Banach space if $\displaystyle Sum||Xn||$ converges then $\displaystyle SumXn$ converges

- Jun 1st 2010, 06:43 AMcizzziBanach space
Can you help me please?

- Jun 1st 2010, 07:02 AMcizzzi
prove that , in a Banach space if $\displaystyle \sum^{\infty}_{n=1}||Xn||$ converges then $\displaystyle \sum^{\infty}_{n=1}Xn$ converges.

- Jun 1st 2010, 07:11 AMFocus
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 $\displaystyle x_n \rightarrow 0$). - Jun 1st 2010, 11:00 AMcizzzi
thank you I try to solve but I do not know very well functional analysis :(

- Jun 1st 2010, 12:55 PMFocus
- Jun 4th 2010, 06:44 AMcizzzi
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? - Jun 6th 2010, 12:22 PMFocus