General forms of the Menshov-Rademacher, Orlicz, and Tandori theorems on orthogonal series - Murach, A. A.; Mikhailets, V. A. (Methods of Functional Analysis and Topology, 2011)
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.