Integral and Differential Equations

In this chapter we consider two of the principal areas of application of the theory of compact operators from Chapter 7. These are the study of integral and differential equations. Integral equations give rise very naturally to compact operators, and so the theory can be applied almost immediately to such equations. On the other hand, as we have seen before, differential equations tend to give rise to unbounded linear transformations, so the theory of compact operators cannot be applied directly. However, with a bit of effort the differential equations can be transformed into certain integral equations whose corresponding linear operators are compact. In effect, we construct compact integral operators by “inverting” the unbounded differential linear transformations, and we apply the theory to these integral operators. Thus, in a sense, the theory of differential equations which we will consider is a consequence of the theory of integral equations. We therefore consider integral equations first.