Let the sequence $u_n\in L^2(0,\infty)$ weakly converges to $u\in L^2(0,\infty)$. What can we say about the corresponding Laplace transforms $u_n(s)$ and $u(s)$?

1. $u_n(s)$ converges point-wise to $u(s)$ for almost all $s>0$.

2. The convergence in (1) but also uniform.