Let X and Y be normed affine spaces and let A be a subset of X and B a subset of Y. A map f: A → B is said to be a smooth homeomorphism if it is a homeomorphism and if each of the maps X ↣ Y; x ⇝ f(x) and is smooth (C1). A map f : X ↣ Y is said to be locally a smooth homeomorphism at a point a ∈ X if there are open neighbourhoods A of a in X and fi of f(a) in Y such that f⊢(A) = B; x ⇝ f(x) and the map is a smooth homeomorphism.

The main theorem of this chapter, the ‘inverse function theorem’, is a criterion for a map f: X ↣ Y to be locally a smooth homeomorphism, when X and Y are complete normed affine spaces. Important corollaries include the ‘implicit function theorem’ and various propositions preliminary to the study of smooth submanifolds. Another corollary is the ‘fundamental theorem of algebra’.

Higher differentials are considered briefly at the end of the chapter.

The increment formula

One of the main tools used in the proof of the inverse function theorem is the ‘increment formula’ (‘la formule des accroissements finis’). This inequality replaces the ‘mean value theorem’, which occurs at this stage in many treatments of the calculus of real-valued functions of one real variable.