Generalized Fourier series in $L^2$ that do not converge pointwise a.e.

For a Hilbert space $L^2$ we have the notion of an orthonormal basis $\{f_j\}$ being a sequence of orthonormal elements such that any element $f$ in $L^2$ can be approximated by partial sums in terms of this basis $$f = \sum_{j=1}^\infty \langle f, f_j \rangle f_j$$ Here the sum converges wrt the $L^2$ norm. This is what I mean by generalized Fourier series.

I have been reading about Carleson's Theorem that says specifically for Fourier series, the series converges pointwise almost everywhere to the approximated function. I have also read that this is not true for a general orthonormal basis. I was hoping someone would be able to provide me with an example demonstrating that statement on a finite measure space, maybe $L^2([0,1])$: A function whose partial sums in terms of the basis do not convergence pointwise almost everywhere.

Thank you for the comments. I was able to use them to provide an answer to When does $\sum_{i=1}^{\infty} X_i$ exist for random sequences $\{X_i\}_{i=1}^{\infty}$?. I'll restate the result from there as it provides an example:

Consider the measure space $L^2([0,1])$ with uniform lebesgue measure. Define the functions (Haar functions) $f_{2^i + k}$ by $$f_{2^i + k} = 2^{i/2}\chi_{\left[\frac{k}{2^i}, \frac{k+1/2}{2^i}\right]} - 2^{i/2}\chi_{\left[\frac{k + 1/2}{2^i}, \frac{k+1}{2^i}\right]}$$ for $i \geq 1, 0 \leq k < 2^i$.

On pg. 598 of the paper mentioned in the comments: Topics in Orthogonal Functions - Price the author states the Haar functions defined above form a complete orthonormal basis. Then pg. 603 of the paper states

For every complete orthonormal system $\Phi$ there is an $L^2$ function $g$ whose $\Phi$-Fourier series can be rearranged to diverge almost everywhere.

So we can finish as follows. Take the Haar functions as above (a complete orthonormal set) and the function $g$ defined above. We write this function's $\Phi$-Fourier series as $$\sum_{n=1}^\infty \langle g, f_n \rangle f_n$$ Now there exists a rearrangement $\sigma$ of the indices of the sum such that it diverges almost everywhere. Now let $g_n = \langle g, f_{\sigma(n)} \rangle f_{\sigma(n)}$ be this rearrangement of terms. The sum $\sum g_n$ diverges almost everywhere.