Properties of expansion systems similar to orthogonal ones

We define expansion systems in a Hilbert space that are similar to orthogonal ones, for which an analogue of Parseval's equality, the extremal property of expansion coefficients, and analogues of the Riesz-Fischer theorem and Bessel's identity (estimating the accuracy of approximation) are valid. In the case when the Hilbert space is the Lebesgue space $L^2$ we prove an analogue of the Men'shov–Rademacher theorem on almost everywhere convergence and analogues of the theorems of Orlicz and Tandori on unconditional convergence. We suggest constructions and examples of non-orthogonal expansion systems similar to orthogonal ones.