Lipschitz Maps and Linearization

Abstract

For the sake of reference we collect together here a more detailed discussion of some of the topics in the text. These topics, Lipschitz functions, the contraction mapping principle, and the Linearization Theorem, were all mentioned briefly in the main body of the text, but in the interest of pursuing other subjects, many instructors will not wish to cover the additional details discussed here. This is particularly true for the proof of the Linearization Theorem, which is perhaps best left to independent study by the students.

As in the proof of the Existence and Uniqueness Theorem, and elsewhere in the text, we find it convenient to use the l ₁ norm

$$||x|| = \sum_{i=1}^n|x_i|,$$

on elements x = (x ₁,…, x _n) of ℝⁿ, because we think the proofs are simpler with this choice. Thus, on elements x in ℝⁿ, the notation ||x|| is used exclusively for the l ₁ norm of x. Other norms on ℝⁿ will be denoted differently. For example |x| = $|x| = ({\sum}_{i=1}^n x^2_i)^{1/2}$ will denote the Euclidean (or l ₂) norm of x and below Theorem B.3 refers to a special norm on ℝⁿ denoted by || ·||0.