  Scattering amplitudes are tempered distributions, which are defined through
their action on functions in the Schwartz space $S(\mathbb{R})$ 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 $S'(\mathbb{R}^+)$. In this
paper, we investigate this problem by characterizing the Mellin transform of
the Schwartz space $S(\mathbb{R}^+)$. 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
$S(\mathbb{R}^+)$. We conclude the paper with applications to tree-level
graviton celestial amplitudes.
