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Surjectivity of the Fourier Transform on Schwartz Space
Consider the Fourier transform on Schwartz space, given by \begin{equation} \mathcal{F}(f)(\xi)= \hat{f}(\xi) = (2\pi)^{-\frac{1}{2}}\int e^{-i\xi x} f(x) \, \mathrm dx \end{equation}
I understand a proof in my notes that shows that we have a left inverse \begin{equation} \mathcal{F}^{-1}(\hat{f})(x) = (2\pi)^{-\frac{1}{2}}\int e^{i\xi x} \hat{f}(\xi) \, \mathrm d\xi \end{equation}
but how do I know that this is also the right inverse?
Many thanks for hints!