Scattering amplitudes are tempered distributions, which are defined through their action on functions in the Schwartz space \( \mathcal{S}\left(\mathbb{R}\right) \) by duality. For massless particles, their conformal properties become manifest when considering their Mellin transform. Therefore we need to mathematically well-define the Mellin transform of distributions in the dual space \( {\mathcal{S}}^{\prime}\left({\mathbb{R}}^{+}\right) \) . In this paper, we investigate this problem by characterizing the Mellin transform of the Schwartz space \( \mathcal{S}\left({\mathbb{R}}^{+}\right) \) . This allows us to rigorously define the Mellin transform of tempered distributions through a Parseval-type relation. Massless celestial amplitudes are then properly defined by taking the Mellin transform of elements in the topological dual of the Schwartz space \( \mathcal{S}\left({\mathbb{R}}^{+}\right) \) . We conclude the paper with applications to tree-level graviton celestial amplitudes.
