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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A note on unconditionally converging series in $L_{p}$
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by Peter Ørno PDF
Proc. Amer. Math. Soc. 59 (1976), 252-254 Request permission


Theorem. A series $\sum {{f_i}}$ in ${L_p}[0, 1](1 \leqslant p \leqslant 2)$ is unconditionally convergent if and only if for each $i$ and for all $t \in [0, 1],\;{f_i}(t) = {\alpha _i}g(t){w_i}(t)$ where $({\alpha _i}) \in {l_2},\;g \in {L_2}[0, 1]$ and $({w_i})$ is an orthonormal sequence in ${L_2}[0, 2]$. This characterization allows the generalization (to u.c. series in ${L_p}[0, 1]$) of several classical theorems concerning almost everywhere convergence of orthogonal series in ${L_2}$.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 59 (1976), 252-254
  • MSC: Primary 46E30; Secondary 40H05
  • DOI:
  • MathSciNet review: 0458156