Convergence of a series in $L^2$ This is probably a very easy exercise or a consequence of some theorem but somehow I am not really sure of the following statement. Suppose $\{f_n\}$ is a sequence of functions in $L^2(\Omega).$ Suppose this is a probability space for now. Define the following function $$g(x)=\sum_{n\in \mathbb{N}} a_n f_n(x)$$
Is it true if $\sum_{n\in \mathbb{N}} |a_n|^2<\infty $ and $f_n\to f$ in $L^2$ then the series converges in $L^2(\Omega)$?
 A: Two conditions under which the series converges in $L^{2}$:
1) $(f_n)$ is orthonormal: in this case you can show that the partial sums  form a Cauchy sequence in $L^{2}$ and use completeness of $L^{2}$
2) $f=0$ and  $\sum \|f_n\|^{2} <\infty$: in this case use the fact that $|\sum_k^{j} a_nf_n(x)|^{2} \leq \sum_k^{j} |a_n|^{2} \sum_k^{j} |f_n(x)|^{2}$.  
A: Here's a condition that various readers may find not so boring:


Suppose $\delta(h)\ge0$ for $h\in\Bbb Z$ and $c^2=\sum_{h\in\Bbb Z}\delta(h)<\infty$. If (on some measure space) $|\int f_j\overline{f_k}|\le\delta(j-k)$ and $s=\sum_{j=-N}^Na_jf_j$ then $||s||_2^2\le c^2\sum|a_j|^2.$
Corollary. If in addition $\sum|a_j|^2<\infty$ then $\sum a_jf_j$ converges in $L^2$.


Note if $(f_j)$ is orthonormal the hypothesis is satisfied with $\delta(0)=1$, $\delta(h)=0$ for $h\ne0$.
Proof: We simplify the notation by regarding $a_j$ as defined for every $j\in\Bbb Z$, with $a_j=0$ for $|j|>N$. Starting from $|s|^2=s\overline s$ and multiplying out the product we obtain $$||s||_2^2=\sum_{j,k}a_j\overline{a_k}\int f_j\overline{f_k}
=\sum_{h\in\Bbb Z}\sum_{j\in\Bbb Z}a_j\overline{a_{j+h}}\int f_j\overline{f_{j+h}}.$$For each $h\in\Bbb Z$ Cauchy-Schwarz implies
$$\left|\sum_{j\in\Bbb Z}a_j\overline{a_{j+h}}\int f_j\overline{f_{j+h}}\right|\le\delta(h)(\sum|a_j|^2)^{1/2}(\sum_j|a_{j+h}|^2)^{1/2}=\delta(h)\sum|a_j|^2.$$
Exercise. Suppose in addition that $||f_j||_2=1$ and $d=\left(\sum_{h\ne0}\delta(h)\right)^{1/2}<1$. Then $||s||_2^2\ge(1-d)^2\sum|a_j|^2$.
I think it's $(1-d)^2$ anyway, may be some other constant.
One can use this to show for example that if $|k_n-n|$ is small enough then $\sum a_je^{ik_jt}$ behaves like a Fourier series...
