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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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