Introduction to Modern Economic Growth two elements of S A metric space is more general than a finite dimensional Euclidean space such as a subset of RK But as with the Euclidean space, we are most interested in defining “functions” from the metric space into itself We will refer to these functions as operators or mappings to distinguish them from real-valued functions Such operators are often denoted by the letter T and standard notation often involves writing T z for the image of a point z ∈ S under T (rather than the more intuitive and familiar T (z)), and using the notation T (Z) when the operator T is applied to a subset Z of S We will use this standard notation here Definition 6.1 Let (S, d) be a metric space and T : S → S be an operator mapping S into itself T is a contraction mapping (with modulus β) if for some β ∈ (0, 1), d(T z1 , T z2 ) ≤ βd(z1 , z2 ), for all z1 , z2 ∈ S In other words, a contraction mapping brings elements of the space S “closer” to each other Example 6.2 Let us take a simple interval of the real line as our space, S = [a, b], with usual metric of this space d(z1 , z2 ) = |z1 − z2 | Then T : S → S is a contraction if for some β ∈ (0, 1), |T z1 − T z2 | ≤ β < 1, |z1 − z2 | all z1 , z2 ∈ S with z1 6= z2 Definition 6.2 A fixed point of T is any element of S satisfying T z = z Recall also that a metric space (S, d) is complete if every Cauchy sequence (whose elements are getting closer) in S converges to an element in S (see the Mathematical Appendix) Despite its simplicity, the following theorem is one of the most powerful results in functional analysis Theorem 6.7 (Contraction Mapping Theorem) Let (S, d) be a complete metric space and suppose that T : S → S is a contraction Then T has a unique fixed point, zˆ, i.e., there exists a unique zˆ ∈ S such that T zˆ = zˆ 267