General forms of the Menshov-Rademacher, Orlicz, and Tandori theorems on orthogonal series

Abstract

We prove that the classical Menshov-Rademacher, Orlicz, and Tandori theorems remain true for orthogonal series given in the direct integrals of measurable collections of Hilbert spaces. In particular, these theorems are true for the spaces L_{2}(X,dμ;H) of vector-valued functions, where (X,μ) is an arbitrary measure space, and H is a real or complex Hilbert space of an arbitrary dimension.

arXiv commentin English, translation of the 1-st (Russian) version, 12 pages