Let $f\_n \in L^2[0,1]$ be an orthonormal sequence and let $c\_n \in \mathbb C$ be such that $\sum\_{n = 1}^{\infty} |c\_n|^2 < \infty$. Does this imply that the sequence $\sum\_{n = 1}^{\infty}c\_nf\_n$ converges pointwise almost everywhere on $[0,1]$?


I guess that this is not true. However, it does hold in some interesting special cases: 


For example, if $f$ is $1$ on $[2k,2k+1)$ and $-1$ on $[2k-1,2k)$ for $k \in \mathbb Z$ and one defines $f\_n(x) = f(2^nx)$, this is true (one proof relies on a theorem about almost sure convergence of independent random variables, see <http://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem>). 


Another interesting case is the choice $f\_n(x) = e^{2\pi i n x}$ where it is true by Carleson's theorem about the almost everywhere convergence of a Fourier series.



A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm
As you remark, this is false in the context of general orthonormal sequence. Generally one has the estimate (the Rademacher-Menshov theorem)


$$|| \max\_{\ell \leq n} |\sum\_{i=1}^{\ell} a\_i \phi\_i| ||\_{L^2} \lesssim \log n (\sum\_{i=1}^{n} |a\_i|^2)^{1/2}$$


where, in general, the factor of $\log n$ is optimal.


Even more surprisingly, perhaps, is that for any complete orthonromal sequence there exists a reordering $\pi$ such that the orthonormal sequence $\{f\_{\pi(n)} \}$ fails to have the almost everywhere convergence property. This is a theorem of Olveskii (see theorem 2 in [Olveskii's book](http://www.ams.org/mathscinet-getitem?mr=470599)).


