I'm trying to understand why there is a continous extenstion of $\mathcal{F}_0f$ to a unitary operator in $L^2$. So let $\mathcal{F}_0f: S(\mathbb{R}^m) \to S(\mathbb{R}^m).$ This is a bijective Operator.
I can show that for $f,g \in \mathcal{S}(\mathbb{R}^m)$ the operator is isometric.
So injectivity follows from isometry, right?
Then I have the claim range($\mathcal{F})=\overline{range \mathcal{F}_0}=L^2(\mathbb{R}^m)$. Why is this true and where can i see this? And is it right that i can follow the surjectivity from this?