Unconditional convergence of general Fourier series

S. Banach, in particular, proved that for any function — even for the constant function $f(x)=1$, where $x\in[0,1]$ — the convergence of its Fourier series with respect to a general orthonormal system (ONS) is not guaranteed.
In this paper, we establish conditions on the functions $\varphi_n$ of an orthonormal system $(\varphi_n)$ under which the Fourier series of functions $f\in\operatorname{Lip}_1$ converges unconditionally almost everywhere.
Our investigation primarily relies on the properties of sequences of linear functionals in Banach spaces, which we use to prove the main results presented in this article. We show that the aforementioned conditions not only exist but are also optimal in a certain sense. Furthermore, we demonstrate that any orthonormal system contains a subsystem for which the Fourier series of every function $f\in\operatorname{Lip}_1$ converges unconditionally.