We consider the problem of extending the concept of bounded-input bounded-output stability to linear time-varying systems with distributional inputs. In particular, the notion of impulse response is examined in a functional analytic setting. This requires that we first extend the classical notion of an integral operator to distribution space. Duality theory for several key normed spaces is then examined. Next, the adjoint operator corresponding to the given system is studied. Finally, necessary and sufficient conditions for stability are established, along with several expressions for the "gain" of the system.