  The primary motivation behind this paper is an attempt to provide a thorough
explanation of how the Mellin transform arises naturally in a process akin to
the construction of the celebrated Gelfand transform. We commence with a study
of a class of Schwartz functions $\mathcal{S}(\mathbb{R}_+),$ where
$\mathbb{R}_+$ is the set of all positive real numbers. Various properties of
this Fr\'echet space are established and what follows is an introduction of the
Mellin convolution operator, which turns $\mathcal{S}(\mathbb{R}_+)$ into a
commutative Fr\'echet algebra. We provide a simple proof of Mellin-Young
convolution inequality and go on to prove that the structure space
$\Delta(\mathcal{S}(\mathbb{R}_+),\star)$ (the space of nonzero, linear,
continuous and multiplicative functionals
$m:\mathcal{S}(\mathbb{R}_+)\longrightarrow \mathbb{R}$) is homeomorphic to
$\mathbb{R}.$ Finally, we show that the Mellin transform arises in a process
which bears a striking resemblance to the construction of the Gelfand
transform.
