Linear maps and impulse responses

Several results are given concerning the representation of causal (mainly) linear operators defined on a set of locally integrable inputs. These results establish the existence of impulse responses and associated integral representations, as well as the convergence of a classical sum to an integral, in a general setting in which a certain continuity condition plays a central role. They also provide necessary and sufficient conditions for a causal linear operator to have a convolution representation with a continuous impulse-response function