Lions and Magens shaked my world.
In the Book of Lions and Magens. They write:
So they define the Fourier transform just this way on $L^2$? How is that possible? I thought this is only possible on $L^1\cap L^2$. Also he defines it first on $L^2$ and then extend it on $\mathcal{S}^\prime$. But shouldn't it be the other way around?
I think that "the integral converging in the sense of $L^2$" means that for $f\in L^2$ we can take a sequence $f_k \in L^1\cap L^2$ that converges to $f$ in $L^2,$ and then $\mathcal{F}f_k$ also converges in $L^2.$ We thus define $\mathcal{F}f$ as this limit.
For your last question: No, it shouldn't be the other way around. $L^2$ is dense in $\mathcal{S}'$ and we can extend the Fourier transform from $L^2$ to $\mathcal{S}'.$