This textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material.
The book covers the main topics of metric space theory that the student of analysis is likely to need. Starting with an overview defining the principal examples of metric spaces in analysis (chapter 1), it turns to the basic theory (chapter 2) covering open and closed sets, convergence, completeness and continuity (including a treatment of continuous linear mappings). There is also a brief dive into general topology, showing how metric spaces fit into a wider theory. The following chapter is devoted to proving the completeness of the classical spaces. The text then embarks on a study of spaces with important special properties. Compact spaces, separable spaces, complete spaces and connected spaces each have a chapter devoted to them. A particular feature of the book is the occasional excursion into analysis. Examples include the Mazur–Ulam theorem, Picard’s theorem on existence of solutions to ordinary differential equations, and space filling curves.
This text will be useful to all undergraduate students of mathematics, especially those who require metric space concepts for topics such as multivariate analysis, differential equations, complex analysis, functional analysis, and topology. It includes a large number of exercises, varying from routine to challenging. The prerequisites are a first course in real analysis of one real variable, an acquaintance with set theory, and some experience with rigorous proofs.
Author(s): Robert Magnus
Series: Springer Undergraduate Mathematics Series
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2022
Language: English
Pages: 244
City: Cham, Switzerland
Tags: Metric Space, Compactness, Separable Space, Completeness, Connectedness, Cantor Set, Banach Fixed Point Theorem, Baire Category Theorem, Ascoli Theorem, Space Filling Curve, Stone-Weierstrass
Preface
Contents
Preliminaries on Sets
Basic Relations
Basic Operations
Writing Predicates
Set-Building Rules
Relations and Functions
Cardinals
Other Notions
1 Metric Spaces
1.1 Metrics
1.1.1 Rationale for Metrics
1.1.2 Defining Metric Space
1.1.3 Exercises
1.2 Examples of Metric Spaces
1.2.1 Normed Spaces
1.2.2 Subspaces
1.2.3 Examples; Not Subspaces of Normed Spaces
1.2.4 Pseudometrics
1.2.5 Cauchy-Schwarz, Hölder, Minkowski
1.2.6 Exercises
1.3 Cantor's Middle Thirds Set
1.3.1 Exercises
1.4 The Normed Spaces of Functional Analysis
1.4.1 Sequence Spaces
1.4.2 Function Spaces
1.4.3 Spaces of Continuous Functions
1.4.4 Spaces of Integrable Functions
1.4.5 Hölder's and Minkowski's Inequalities for Integrals
1.4.6 Exercises
2 Basic Theory of Metric Spaces
2.1 Balls in a Metric Space
2.1.1 Limit of a Convergent Sequence
2.1.2 Uniqueness of the Limit
2.1.3 Neighbourhoods
2.1.4 Bounded Sets
2.1.5 Completeness; a Key Concept
2.1.6 Exercises
2.2 Open Sets, and Closed
2.2.1 Open Sets
2.2.2 Union and Intersection of Open Sets
2.2.3 Closed Sets
2.2.4 Union and Intersection of Closed Sets
2.2.5 Characterisation of Open and Closed Sets by Sequences
2.2.6 Interior, Closure and Boundary
2.2.7 Limit Points of Sets
2.2.8 Characterisation of Closure by Limit Points
2.2.9 Subspaces
2.2.10 Open and Closed Sets in a Subspace
2.2.11 Exercises
2.3 Continuous Mappings
2.3.1 Defining Continuity
2.3.2 New Views of Continuity
2.3.3 Limits of Functions
2.3.4 Characterising Continuity by Sequences
2.3.5 Lipschitz Mappings
2.3.6 Examples of Continuous Functions
2.3.7 Exercises
2.4 Continuity of Linear Mappings
2.4.1 Continuity Criterion
2.4.2 Operator Norms
2.4.3 Exercises
2.5 Homeomorphisms and Topological Properties
2.5.1 Equivalent Metrics
2.5.2 Exercises
2.6 Topologies and σ-Algebras
2.6.1 Order Topologies
2.6.2 Exercises
2.6.3 Pointers to Further Study
2.7 () Mazur-Ulam
2.7.1 Exercises
3 Completeness of the Classical Spaces
3.1 Coordinate Spaces and Normed Sequence Spaces
3.1.1 Completeness of Rn
3.1.2 Completeness of p
3.1.3 Exercises
3.2 Product Spaces
3.2.1 Finitely Many Factors
3.2.2 Infinitely Many Factors
3.2.3 The Space 2N+ and the Cantor Set
3.2.4 Subspaces of Complete Spaces
3.2.5 Exercises
3.3 Spaces of Continuous Functions
3.3.1 Uniform Convergence
3.3.2 Series in Normed Spaces
3.3.3 The Weierstrass M-Test
3.3.4 The Spaces C(R) and Cp(R)
3.3.5 Exercises
3.4 () Rearrangements
3.4.1 Vector Series
3.4.2 Exercises
3.4.3 Pointers to Further Study
3.5 () Invertible Operators
3.5.1 Fredholm Integral Equation
3.5.2 Exercises
3.5.3 Pointers to Further Study
3.6 () Tietze
3.6.1 Formulas for an Extension
3.6.2 Exercises
3.6.3 Pointers to Further Study
4 Compact Spaces
4.1 Sequentially Compact Spaces
4.1.1 Continuous Functions on Sequentially Compact Spaces
4.1.2 Bolzano-Weierstrass in Rn
4.1.3 Sequentially Compact Sets in Rn
4.1.4 Sequentially Compact Sets in Other Spaces
4.1.5 The Space C(M)
4.1.6 Exercises
4.2 The Correct Definition of Compactness
4.2.1 Thoughts About the Definition
4.2.2 Compact Spaces and Compact Sets
4.2.3 Continuous Functions on Compact Spaces
4.2.4 Uniform Continuity
4.2.5 Exercises
4.3 Equivalence of Compactness and Sequential Compactness
4.3.1 Relative Compactness
4.3.2 Local Compactness
4.3.3 Exercises
4.4 Finite Dimensional Normed Vector Spaces
4.4.1 Exercises
4.5 () Ascoli
4.5.1 Peano's Existence Theorem
4.5.2 Exercises
4.5.3 Pointers to Further Study
5 Separable Spaces
5.1 Dense Subsets of a Metric Space
5.1.1 Defining a Vector-Valued Integral
5.1.2 Exercises
5.2 Separability
5.2.1 Second Countability
5.2.2 Exercises
5.3 () Weierstrass
5.3.1 Exercises
5.3.2 Pointers to Further Study
5.4 () Stone-Weierstrass
5.4.1 Exercises
5.4.2 Pointers to Further Study
6 Properties of Complete Spaces
6.1 Cantor's Nested Intersection Theorem
Notes About Cantor's Theorem
6.1.1 Categories
Thoughts About the Proof
6.1.2 Exercises
6.2 () Genericity
6.2.1 Exercises
6.2.2 Pointers to Further Study
6.3 () Nowhere Differentiability
6.3.1 Exercises
6.3.2 Pointers to Further Study
6.4 Fixed Points
6.4.1 Exercises
6.5 () Picard
6.5.1 Exercises
6.6 () Zeros
6.6.1 Exercises
6.6.2 Pointers to Further Study
6.7 Completion of a Metric Space
6.7.1 Other Ways to Complete a Metric Space
6.7.2 Exercises
7 Connected Spaces
7.1 Connectedness
7.1.1 Connected Sets
7.1.2 Rules for Connected Sets
7.1.3 Connected Subsets of R
7.1.4 Exercises
7.2 Continuous Mappings and Connectedness
7.2.1 Continuous Curves
7.2.2 Arcwise Connectedness
7.2.3 Exiting a Set
7.2.4 Exercises
7.3 Connected Components
7.3.1 Examples of Connected Components
7.3.2 Arcwise Connected Components
7.3.3 Exercises
7.4 () Peano
7.4.1 Exercises
7.4.2 Pointers to Further Study
Afterword
Index
