Operators, in particular linear operators, are the key focus of functional analysis. Operators connect function spaces with each other, describe properties of physical systems, describe cost functions, and solve partial differential equations (PDEs). This text is mostly concerned with linear operators on Banach spaces with one exception in the chapter on Fixed-Point theorems (Chapter 7), where we also consider fixed points of non-linear operators. During this chapter, and then continuing through the entire text, we will see four types of operators as standard examples: (i) a matrix \(A\) on \(\RR^n\text{,}\) which is used to relate the abstract theory back to what we know from linear algebra; (ii) the Laplacian operator \(\Delta\text{,}\) is of central importance for the analysis of PDEs, and it is and unbounded linear operator with its own challenges; (iii) an integral operator, which is compact. Compactness will, of course, make many arguments much easier. Finally, (iv) the first derivative \(\frac{\partial}{\partial x}\) arises as an unbounded generator of the shift semigroup (see Chapter 9).
