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Let $X$ be a Banach space. A series $\sum x_n$ in $X$ is weakly unconditionally Cauchy (or weakly uncontionally convergent) if $\sum |x^*(x_n)| < +\infty$ for every $x^* \in X^*$.

Exercise 3, page 114 of Diestel's Sequences and Series in Banach Spaces reads:

The bounded linear operators from a Banach space $X$ into $\ell_1$ correspond precisely to the sequences $(x_n^*)$ in $X^*$ for which $\sum |x^*_n(x)| < +\infty$, for each $x \in X$, i. e., the weakly unconditionally Cauchy series in $X^*$.

I'm having trouble with the bolded part. More precisely, I don't see why $\sum |x^*_n(x)| < +\infty$ for each $x \in X$ implies that $\sum x^*_n$ is weakly unconditionally Cauchy in $X^*$ (that is, $\sum |x^{**} (x^*_n)| < +\infty$ for every $x^{**} \in X^{**}$).

Thanks in advance :)

Vinícius Morelli
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    I may be off here, but: a given $(x_n^)$ defines a bounded operator taking $x$ to $( x_n^x)$. Proving the other direction for the adjoint of this particular operator restricted to $c_0$ (use exercise 2) shows $(x_n^*)$ is in fact WuC. – David Mitra Nov 25 '20 at 18:26
  • That settles it! Thanks a lot. – Vinícius Morelli Nov 25 '20 at 21:04

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