Unbounded Linear Operators

In 4.2.3 we defined a bounded linear operator in the setting of two normed linear spaces E x and E y and studied several interesting properties bounded linear operators. But if said operator ceases to be bounded, then we get an unbounded linear operator. The class of unbounded linear operators include a rich class of operators, notably the class of differential operators. In 4.2.11 we gave an example of an unbounded differential operator. There are usually two different approaches to treating a differential operator in the usual function space setting. The first is to define a new topology on the space so that the differential operators are continuous on a nonnormable topological linear space. This is known as L. Schwartz's theory of distribution (Schwartz, [52]). The other approach is to retain the Banach space structure while developing and applying the general theory of unbounded linear operators (Browder, F [9]). We will use the second approach. We have already introduced closed operators in Chapter 7. The linear differential operators are usually closed operators, or at least have closed linear extensions. Closed linear operators and continuous linear operators have some common features in that many theorems which hold true for continuous linear operators are also true for closed linear operators. In this chapter we point out some salient features of the class of unbounded linear operators.