Most of us know from distribution theory that a good understanding of certain operators requires to make use, aside from the Hilbert space ${\mathcal H} = L^2(R^d)$, of certain spaces of generalized functions. A standard approach is based on the fact that one defines distributions as members of the dual space a space of test functions, which is dense in ${\mathcal H}$. We will show that time-frequency analysis allows the introduction of a Banach spaces $S_0(R^d)$ which is suitable as an algebra of test functions, dense in ${\mathcal H}$. Together with its dual it forms a Gelfand triple of Banach spaces. The Fourier transform is an example: At the level of $S_0$ it is well defined as an integral transform (also the inverse Fourier transform), at the $L^2$-level it is unitary, while one can express at the level distributions that "pure frequencies" are mapped onto Diracs, and that this fact characterizes the Fourier transform as Gelfand tripe isomorphism. Similarly one can view the mapping between the (integral) kernels describing linear mappings and their spreading symbol (a distribution over phase space) as a Gelfand triple isomorphism, characterized by the fact that pure time-frequency shifts are mapped onto corresponding Dirac measures.