The space of multipliers of S' and the S-families of tempered distributions

In this paper we study two operations introduced by D. Carfì in the sets L(Sn; Sm) and SL(m; S0n ); the last space was deeply studied in [CG]. In particular we establish some their algebraic and analytic properties. The new concepts are: 1) product of an operator in L(Sn; Sm) by an OM function; 2) product of a family in S(m; S0n ) by an OM function. These new operations permit the development of a generalization of linear algebra in the space of tempered distributions and a more deeply study of some problems faced by linear algebra, as the theory of system, theory of decision, the optimal control, the quantum mechanics and so on.