An Illustrative Introduction to Modern Analysis 1st Edition by Nikolaos Katzourakis 9781138718272 1138718270

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ISBN 10: 1138718270
ISBN 13: 9781138718272
Author: Nikolaos Katzourakis

Aimed primarily at undergraduate level university students, An Illustrative Introduction to Modern Analysis provides an accessible and lucid contemporary account of the fundamental principles of Mathematical Analysis. The themes treated include Metric Spaces, General Topology, Continuity, Completeness, Compactness, Measure Theory, Integration, Lebesgue Spaces, Hilbert Spaces, Banach Spaces, Linear Operators, Weak and Weak* Topologies. Suitable both for classroom use and independent reading, this book is ideal preparation for further study in research areas where a broad mathematical toolbox is required

An Illustrative Introduction to Modern Analysis 1st Table of contents:

1. Sets, Mappings, Countability and Choice

   • Prerequisites for the Reading of This Book
   • A Quick Review of Sets and Mappings
   • Cardinality and Countability of Sets
   • The Axiom of Choice and Zorn’s Lemma
   • Literally Every Vector Space Has a Basis
   • Exercises

2. Metric Spaces and Normed Spaces

   • Abstracting the Concepts of Analysis
   • Metrics on Sets and Norms on Vector Spaces
   • Bounded Sets, Convergence and Continuity
   • Balls, Open Sets and Closed Sets
   • Metric Topologies and Equivalent Metrics
   • Metric and Normed Subspaces
   • Interior, Closure and Boundary of a Set
   • Some Concrete Examples of Normed Spaces
     • The Euclidean Space
     • The Class of ℓp Spaces
     • Spaces of Continuous Functions
   • Separable Metric Spaces and the ℓp Class
   • Exercises

3. Completeness and Applications

   • The Significance of Completeness
   • Complete Metric Spaces and Banach Spaces
   • Cantor’s Intersection and Baire’s Category Theorems
   • Series in Normed Vector Spaces
   • Are Our Favourite Normed Spaces Actually Complete?
     • Completeness of the ℓp Spaces
     • Spaces of Bounded and Continuous Functions
   • The Banach Fixed-Point Theorem
   • An Application to Differential Equations
   • Exercises

4. Topological Spaces and Continuity

   • From Metric to Topological Spaces and Beyond
   • The Endgame of Abstraction of Analysis
   • Open Sets and Closed Sets
   • Topological Subspaces
   • Convergence and Continuity
   • Interior, Closure and Boundary of a Set
   • Subbases, Weak and Product Topology
   • Exercises

5. Compactness and Sequential Compactness

   • Compactness Beyond Euclidean Spaces
   • The Class of Sequentially Compact Sets
   • The Class of Compact Sets
   • Compactness and Finite Dimensionality in Normed Spaces
   • Continuous and Uniformly Continuous Maps
   • Compact Sets in the Space of Continuous Functions
   • Exercises

6. The Lebesgue Measure on the Euclidean Space

   • Transcending Lengths, Areas and Volumes
   • The Pathway to Measuring Sets on ℝd
   • Elementary Sets and Content
   • The Lebesgue Outer Measure
   • Measurable and Non-Measurable Sets
   • The Carathéodory Theorem
   • Exercises

7. Measure Theory on General Spaces

   • Measuring Beyond the Euclidean Setting
   • Measurable Spaces and Measure Spaces
   • Outer Measures and Carathéodory’s Theorem
   • Continuity and Other Properties of Measures
   • Measurable Functions and Mappings
   • Simple Functions and Mappings
   • Exercises

8. The Lebesgue Integration Theory

   • From Riemann’s to Lebesgue’s Integral
   • Integration of Simple Functions
   • Integration of Non-Negative Measurable Functions
   • The Monotone Convergence Theorem
   • Almost Everywhere Properties and Fatou’s Lemma
   • Integration of Measurable Maps and the Space L1
   • The Dominated Convergence Theorem in L1
   • Almost Uniform Convergence and Egoroff’s Theorem
   • The Riemann Versus the Lebesgue Integral
   • Product Measures and the Fubini-Tonelli Theorem
   • The Linear Change of Variables Formula
   • Exercises

9. The Class of Lebesgue Functional Spaces

   • From Integration to Functional Spaces
   • The Lp Spaces: Definition and First Properties
   • The Inequalities of Hölder and Minkowski
   • Completeness and Density of Simple Maps
   • Density of C0 and Separability in the Euclidean Case
   • Mollification and Subspaces of Smooth Functions
     • Convolution and Young Inequality
     • Mollification by Convolution
   • Exercises

10. Inner Product Spaces and Hilbert Spaces

• Euclidean Structures Past the Euclidean Realm
• Inner Products on Vector Spaces
• Angles, Orthogonality and Orthogonal Splittings
• Orthonormal Sets and Orthonormal Bases
• Isometries and Classification of Hilbert Spaces
• Exercises

11. Linear Operators on Normed Spaces

• What Is Functional Analysis All About
• Bounded Linear Operators on Normed Spaces
• The Banach-Steinhaus and Open Mapping Theorems
• Dual Spaces and the Riesz Representation Theorem
• The Analytic Hahn-Banach Extension Theorems
• The Geometric Hahn-Banach Separation Theorems
• Exercises

12. Weak Topologies on Banach Spaces

• The Raison d’Être for Weakening Topologies
• The Weak Topology of a Banach Space
• On the Nature of Weakly Open Sets
• The Mazur Theorem for Convex Sets
• Weak Convergence and Its Properties
• Weak Lower Semi-Continuity and Convexity
• Exercises

13. Weak Topologies and Compactness*

• Weak Topologies Are Not Weak Enough
• The Weak* Topology of a Dual Banach Space
• On the Nature of Weakly* Open Sets
• Weak* Convergence and Its Properties
• Weak* Compactness in Dual Banach Spaces
• Weak Compactness in Reflexive Banach Spaces
• Uniformly Convex Banach Spaces
• Exercises

14. Functional Properties of the Lebesgue Spaces

• The Lebesgue Spaces Are Banach Spaces After All
• Uniform Convexity and Reflexivity for 1 < p < ∞
• Identifying the Duals of the Lebesgue Spaces
• The True Meaning of Weak Convergence in Lp
• Sequential Weak Compactness and Weak* Compactness
• Exercises

15. Solutions to the Exercises

• Solutions to the Exercises in Chapter 1
• Solutions to the Exercises in Chapter 2
• Solutions to the Exercises in Chapter 3
• Solutions to the Exercises in Chapter 4
• Solutions to the Exercises in Chapter 5
• Solutions to the Exercises in Chapter 6
• Solutions to the Exercises in Chapter 7
• Solutions to the Exercises in Chapter 8
• Solutions to the Exercises in Chapter 9
• Solutions to the Exercises in Chapter 10
• Solutions to the Exercises in Chapter 11
• Solutions to the Exercises in Chapter 12
• Solutions to the Exercises in Chapter 13
• Solutions to the Exercises in Chapter 14

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