Differential Calculus in Banach Spaces

The differential calculus is one of the fundamental techniques of nonlinear functional analysis. Very often we will use this notion. In this chapter, we develop the calculus in real Banach spaces. Section 3.1 deals with definitions on Gâteaux and Frechat derivative with illustrative examples. We also give a variant of mean value theorem. Properties of the derivative are discussed in Sect. 3.2, while in Sect. 3.3, we discuss partial derivatives. Section 3.4 deals with higher derivative. Subsequently, we give Taylor’s theorem, inverse function and implicit function theorems.