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 $L_{2}(X,d\mu;H)$ of vector-valued functions, where $(X,\mu)$ is an arbitrary measure space, and $H$ is a real or complex Hilbert space of an arbitrary dimension.