Let $$f_i=\exp(\int_0^T h_i(s)\,{\rm d}W_s-1/2\int_0^T h^2_i(s)\,{\rm d}s)$$ where $W_s$ is a brownian motion in a probability space $(\Omega,F,P) $ and $h_i\in L^2(0,T) $. Suppose $F_n\to F$ in $L^2(\Omega)$ where $F_n$ is a finite linear combination of functions $f_j$ (that is $F_n=\sum_{j=1}^{k_n}a_i f_{i_j}$). I want to prove that $$G_n(\omega,t)=\sum_{j=1}^{k_n}a_i f_{i_j}(\omega)h_{i_j}(t)$$ converges in $L^2(\Omega\times (0,T))$.
I have managed to prove it only for $F_n=f_i$ but I can't extend it to linear combinations. I would appreciate any possible idea. Thank you in advance.