Banach space

**cizzzi** · June 1st 2010, 06:42 AM

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

**cizzzi** · June 1st 2010, 06:43 AM

Can you help me please?

**cizzzi** · June 1st 2010, 07:02 AM

prove that , in a Banach space if $\sum^{\infty}_{n=1}||Xn||$ converges then $\sum^{\infty}_{n=1}Xn$ converges.

**Focus** · June 1st 2010, 07:11 AM

Originally Posted by cizzzi

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 $x_n \rightarrow 0$ ).

**cizzzi** · June 1st 2010, 11:00 AM

thank you I try to solve but I do not know very well functional analysis

**Focus** · June 1st 2010, 12:55 PM

Originally Posted by cizzzi

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)?

**cizzzi** · June 4th 2010, 06:44 AM

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?

**Focus** · June 6th 2010, 12:22 PM

Originally Posted by cizzzi

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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