Okay, so I'm trying to understand the proof of the classical CLT. I'm not really a probability guy, so I'll apologize in advance for the analysis-heavy language used.
Statement: If $X_i\in L^1\cap L^2$ independent with constant mean and second moment. Then $$ \sum_{i=1}^N\frac{x_i}{\sqrt{N}} $$ converges weakly (in the probabilist sense) to a Gaussian.
Looking at the proof, we show essentially that the Fourier Transform (characteristic function) of a sequence of functions (rvs) $f_n$ converges to the Fourier Transform of a Gaussian at least pointwise.
Now, what confuses me is how we complete the proof. We know that the Fourier Transform is an Isometry on $L^2$, so why don't we use this? The standard proof seems to go through either some sort of approximation-theoretic argument using essentially Schwartz class functions and their density to obtain weak convergence or via something called the continuity theorem. Isn't there a another/easier way using the Fourier Isometry (of $L^2$)? The idea here is of course that isometry somehow means that both spaces have the same topology and thus convergence in either is equivalent to convergence in the other.
I'd appreciate any arguments as to why replacing Schwartz argument with the $L^2$ argument does or does not work! References are of course also appreciated.