Let $\{f_{\alpha_n}\}\subset{\cal L}_2^0(\mathbb R)$ be a sequence function converging to $g$ where ${\cal L}_2^0(\mathbb R)$ is a Banach space defined by $$ {\cal L}_2^0(\mathbb R)=\left\{h:\int_{\mathbb R}h=0\textrm{ and } \int_\mathbb Rh^2<\infty\right\} $$ with norm $\|h\|=\left(\int_\mathbb Rh^2\right)^{\frac12}.$
Now, suppose that I am given $\{\alpha_n\}$ is a sequence of function in ${\cal C}^2$ converging to $\alpha_0$.
I wish to have $f_{\alpha_0}=g.$ Is it correct? I have this conjecture due to the uniqueness of limit, i.e., since $\{f_{\alpha_n}\}\to f_{\alpha_0}$. However, I haven't a clue to prove analytically. Any suggestion?
I believe the extra conditions to your problem are unnecessary, and what you are essentially asking is if the $ L^2$ limit is the same as the $ C^2$ limit.
I claim they are the same for a $ C^0$ limit. To do so you only need to show that their $ L^2 $ difference is 0 on any compact set $ K $. Compare this $ L^2$ difference to the $ L^2$ differences between each limit and the sequence. Then use that $ K $ is compact to get that the terms coming from $ C^0$ limit approach 0. The $ L^2$ convergence will give you that the other terms approach 0.
So you get that the limits are the same for any compact set. Hence they are equal.