Calculus in Banach Spaces

Abstract

In this chapter we develop the theory of the derivative for mappings between Banach spaces. Partial derivatives, Jacobians, and gradients are all examples of the general theory, as are the Gâteaux and Fréchet differentials. Kantorovich’s theorem on Newton’s method is proved. Following that there is a section on implicit function theorems in a general setting. Such theorems can often be used to prove the existence of solutions to integral equations and other similar problems. Another section, devoted to extremum problems, illustrates how the methods of calculus (in Banach spaces) can lead to solutions. A section on the “calculus of variations” closes the chapter.