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Table of Contents
Intro
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
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
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
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