Introductory Linear Operator Theory

In this chapter we apply concepts of functional analysis, especially those concepts related to Hilbert and Banach spaces, to introduce basic operator theory relevant to applied electromagnetics. We begin with the definition of a linear operator and provide examples of common operators that arise in physical problems. We next define linear functionals as a special class of linear operators. Linear functionals occur quite often in electromagnetics and are very useful in theoretical investigations and in formulating problems to be solved numerically. In addition, the concept of a linear functional, in conjunction with the Riesz representation theorem, gives an appropriate motivation for introducing the important concept of an adjoint operator. Next, the class of self-adjoint operators is discussed, as well as the broader category of normal operators. We will see later that self-adjoint operators, and especially compact self-adjoint operators, have very nice mathematical properties that can be usefully exploited. Definite operators are then discussed, which themselves are contained within the class of self-adjoint or symmetric operators, and lead to a new notion of convergence. Compact operators are introduced, both at the function and sequence (infinite matrix) levels, and examples from applied mathematics and electromagnetics are provided.