A function space approach to the foundations of system theory

The idea of using input-output relations to define abstract systems is criticized to the effect that the arbitrariness in state space association it generates does not relate to its practical use for control problems. This prompts the exploration of different points of view and here input-output function spaces take the place of input-output relations. The theory takes profoundly different directions according to the mathematics into which it is introduced. In this paper, after laying some foundations in rather general settings, a crucial part is devoted to a Hilbert space setting and discusses for linear systems the completion of the spaces of free responses, the relation with semigroup theory, and the construction of linear topological spaces of linear systems.