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On a theorem of Kwapień. (English) Zbl 1019.46010
Summary: If $$(\Omega,\Sigma,\nu)$$ is the Cantor group with its resident Haar measure, $$X$$ a Banach space and $$\{x_n\}$$ a sequence in $$X$$, a classic theorem of Kwapien is used in this note to investigate the relationship between the convergence of the series $$\sum^\infty_{n=1} \omega_nx_n$$ in $$L_1 (\nu,X)$$ and the existence of certain convergent subseries of $$\sum^\infty_{n=1} x_n$$ in $$X$$.
MSC:
 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46B09 Probabilistic methods in Banach space theory
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References:
 [1] Diestel J., Sequences and series in Banach spaces 92 (1984) · Zbl 0542.46007 [2] Halmos P. R., Measure Theory 18 (1950) · Zbl 0040.16802 [3] Hoffmann-Jørgensen J., Studia Math. 52 pp 159– (1974) [4] Kwapień S., Studia Math. 52 pp 187– (1974) [5] Vakhania N. N., Probability distributions on Banach spaces 14 (1987)
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