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Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with

$$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$

which means there exists no $x \in X$ with

$$\lim_{n \to \infty} \left\|x-\sum_{k=1}^{n} x_k \right\|=0. $$

Can anybody give me an example of such a sequence $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}}$?

I know that this situation is only possible if $X$ is not complete.

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If $(x_n)$ is Cauchy, it has a subsequence $(y_n)$ with $\Vert y_{n+1}-y_n\Vert<1/2^n$. Consider the series $y_1+(y_2-y_1)+(y_3-y_2)+\cdots$. It converges if and only if $(y_n)$ is convergent. – David Mitra Jul 19 '14 at 0:07

Take $X=c_{00}$---the space of all sequences which are almost everywhere $0$ and as $x_n$---the sequence having $\frac{1}{2^n}$ on $n$-th place and $0$ elsewhere.

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So we get $\sum_{n=1}^{\infty} \|x_n\|_1=\sum_{n=1}^{\infty} 2^{-n}=1$ and $\sum_{n=1}^{\infty} x_n = \left\lbrace 2^{-n} \right\rbrace_{n \in \mathbb{N}} \notin X $. – user152546236 Jul 19 '14 at 0:57

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