We will indicate how the theory of Banach Gelfand Triples (a variant of the so-called rigged Hilbert spaces used in quantum mechanics), based on the Segal algebar $S_0(R^d)$ can be used to give a meaning to the expressions arising in Dirac's calculus, using the bra-kets. The Fourier transform is the perfect example, showing that one cannot stay within the Hilbert space setting of $L^2(R^d)$, because its building blocks -S the pure frequencies -are not square integrable. Moreover they form a continuously parametrized family (however not a continuous frame). The kernel theorem as well as the composition law - described at the kernel level - gives a way to reinterpret some of the results given by Dirac as statements about the composition of (unitary) Banach Gelfand Triple isomorphisms. In this way also the notational connection between the Kronecker $\delta$-symbol and the Dirac symbols can be pointed out in a natural way. In fact, from our point of view it is a way to describe the analogue of unitary matrices $U$ over $C^n$, which have the property that $ U' * U = Id_n = U * U'$ , where $U'$ describes the adjoint (transpose + conjugate) matrix. Special emphasis is put on the relevance of $w^*$-convergence within the dual space of $S_0(R^d)$.