Certain properties of multi-input, multi-output dynamic systems which are of interest in automatic control are investigated. It is shown that such systems can be viewed as a mapping whose domain is the set of possible input functions and whose range is either the set of responses or a set of equivalence classes. By selecting the norms for the input and output spaces in various ways it is possible to interpry of the familiar properties of a system in terms of these mappings and new means of system characterization are suggested as well. It is shown that the study of these mappings leads naturally to the study of an inverse equation. Conditions under which the inverse equation exists are derived for some linear and nonlinear systems and explicit representations for the inverse equation are given for certain classes of linear systems. The effect of feedback is analyzed in terms of its influence on the character of these mappings and certain limitations on feedback as a device for altering a given system are noted. It is also shown that the inverse equation can be used to help define the optimum input for a certain type of time-optimal problems where the objective is stated in terms of the outputs rather than the state.