Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgments
- Notations and Abbreviations
- 1 Basic Definitions and Concepts from Metric Spaces
- 2 Fixed Point Theory in Metric Spaces
- 3 Set-valued Analysis: Continuity and Fixed Point Theory
- 4 Variational Principles and Their Applications
- 5 Equilibrium Problems and Extended Ekeland’s Variational Principle
- 6 Some Applications of Fixed Point Theory
- Appendix A Some Basic Concepts and Inequalities
- Appendix B Partial Ordering
- References
- Index
1 - Basic Definitions and Concepts from Metric Spaces
Published online by Cambridge University Press: 15 July 2023
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgments
- Notations and Abbreviations
- 1 Basic Definitions and Concepts from Metric Spaces
- 2 Fixed Point Theory in Metric Spaces
- 3 Set-valued Analysis: Continuity and Fixed Point Theory
- 4 Variational Principles and Their Applications
- 5 Equilibrium Problems and Extended Ekeland’s Variational Principle
- 6 Some Applications of Fixed Point Theory
- Appendix A Some Basic Concepts and Inequalities
- Appendix B Partial Ordering
- References
- Index
Summary
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,
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- Publisher: Cambridge University PressPrint publication year: 2023