In this chapter, we gather some basic definitions, concepts, and results from metric spaces which

are required throughout the book. For detail study of metric spaces, we refer to [8, 46, 61, 95, 110,

150, 154].

Definitions and Examples

Definition 1.1 Let X be a nonempty set. A real-valued function dX × X →is said to be a

metric on X if it satisfies the following conditions:

The set X together with a metric d on X is called a metric space and it is denoted by (X, d). If there

is no confusion likely to occur we, sometime, denote the metric space (X, d) by X.

Example 1.1 Let X be a nonempty set. For any x, y ∈ X, define

Then d is a metric, and it is called a discrete metric. The space (X, d) is called a discrete metric space.

The above example shows that on each nonempty set, at least one metric that is a discrete metric

can be defined.

Example 1.2 Let X = n, the set of ordered n-tuples of real numbers. For any x = (x1, x2, … , xn) ∈

X and y = (y1, y2, … , yn) ∈ X, we define

Then, d1, d2, dp (p ≥ 1), d∞ are metrics on n.

Example 1.3 Let ℓ∞ be the space of all bounded sequences of real or complex numbers, that is,

is a metric on ℓ∞ and (ℓ∞, d∞) is a metric space.

Example 1.4 Let s be the space of all sequences of real or complex numbers, that is,

is a metric on s.

Example 1.5 Let ℓp, 1 ≤ p < ∞, denote the space of all sequences ﹛xn﹜ of real or complex numbers such that that is,

is a metric on ℓp and (ℓp, d) is a metric space.

Example 1.6 Let B[a, b] be the space of all bounded real-valued functions defined on [a, b], that is,