Economic growth and economic development 281

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.
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