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Characterizations of series in Banach spaces. (English) Zbl 0952.46009
The authors of this paper consider weakly unconditionally Cauchy series in Banach spaces. It is well known that:
1. A weakly unconditionally Cauchy series in a Banach space can be characterized as a series $$\sum_i x_i$$ such that, for every null sequence $$(t_i)_i$$, $$\sum_i t_ix_i$$ is convergent.
2. In a normed space $$X$$, $$\sum^\infty_{i=1} x_i$$ is a weakly unconditionally Cauchy series if and only if the set $E= \Biggl\{ \sum^n_{i=1} \alpha_i x_i:n\in\mathbb{N},\;|\alpha_i|\leq 1, i\in\{1,\dots, n\}\Biggr\}$ is bounded.
3. If $$X$$ is a Banach space then the following conditions are equivalent:
(a) There exists a weakly unconditionally Cauchy series which is convergent, but is it not unconditionally convergent, in $$X$$.
(b) There exists a weakly unconditionally Cauchy series which is weakly convergent, but is not convergent.
(c) There exists a weakly unconditionally Cauchy series which is not weakly convergent.
(d) The space $$X$$ has a copy of $$c_0$$.
For any given series $$\zeta= \sum_i x_i$$ in $$X$$ consider the sets.
1. $${\mathcal S}={\mathcal S}(\zeta)$$, of sequences $$(a_i)_i\in\ell_\infty$$ such that $$\sum_i a_ix_i$$ converges.
2. $${\mathcal S}_w={\mathcal S}_w(\zeta)$$, of sequences $$(a_i)_i\in\ell_\infty$$ such that $$\sum_i a_ix_i$$ is weakly convergent.
3. $${\mathcal S}_0={\mathcal S}_0(\zeta)$$, of sequences $$(a_i)_i\in\ell_\infty$$ such that $$\sum_i a_ix_i$$ is $$*$$-weakly convergent (i.e. $$\left(\sum^n_{i= 1}a_ix_i\right)_n$$ converges with respect to the topology $$\sigma(X^{**},X^*))$$.
These sets, endowed with the sup norm, will be called the spaces of convergence, of weakly convergence and of weak-$$*$$ convergence of the series $$\zeta$$, respectively.
It is clear that if the Banach space $$X$$ does not have a copy of $$c_0$$, then every weakly unconditionally Cauchy series is unconditionally convergent and we have $${\mathcal S}={\mathcal S}_w={\mathcal S}_0= \ell_\infty$$. The authors obtain the following
Theorem. Let $$\sum_i x_i$$ be a series in a Banach space $$X$$. The following statements are equivalent:
1. $${\mathcal S}_0= \ell_\infty$$.
2. The space $${\mathcal S}_0$$ is complete.
3. The series $$\sum_i x_i$$ is a weakly unconditionally Cauchy series.
Using sequence spaces of reals such as $${\mathcal S}$$, $${\mathcal S}_w$$, $${\mathcal S}_0$$, they considered some characterization of unconditionally Cauchy series in a Banach space.

##### MSC:
 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46B45 Banach sequence spaces 40A05 Convergence and divergence of series and sequences
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