ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 68,   2   (1999)
pp.   337-344

CHARACTERIZATIONS OF SERIES IN BANACH SPACES
A. AIZPURU and F. J. PEREZ-FERNANDEZ

Abstract.  In this paper we prove several new characterizations of weakly unconditionally Cauchy series in Banach spaces and in the dual space of a normed space. For a given series $\zeta$, we consider the spaces $\mathcalS(\zeta)$, $\mathcalS_w(\zeta)$ and $\mathcalS_0(\zeta)$ of bounded sequences of real numbers $(a_i)_i$ such that the series $\sum_i^a_ix_i$ is convergent, weakly convergent or $\ast$-weakly convergent, respectively. By means of these spaces we characterize conditionally and weakly unconditionally Cauchy series.

AMS subject classification.  46B15, 46B99; Secondary 40A05, 46B45
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