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Mathematical Analysis Fundamentals

Book Description

The author’s goal is a rigorous presentation of the fundamentals of analysis, starting from elementary level and moving to the advanced coursework. The curriculum of all mathematics (pure or applied) and physics programs include a compulsory course in mathematical analysis. This book will serve as can serve a main textbook of such (one semester) courses. The book can also serve as additional reading for such courses as real analysis, functional analysis, harmonic analysis etc. For non-math major students requiring math beyond calculus, this is a more friendly approach than many math-centric options.



  • Friendly and well-rounded presentation of pre-analysis topics such as sets, proof techniques and systems of numbers.
  • Deeper discussion of the basic concept of convergence for the system of real numbers, pointing out its specific features, and for metric spaces

  • Presentation of Riemann integration and its place in the whole integration theory for single variable, including the Kurzweil-Henstock integration

  • Elements of multiplicative calculus aiming to demonstrate the non-absoluteness of Newtonian calculus.

Table of Contents

  1. Cover image
  2. Title page
  3. Copyright
  4. Dedication
  5. Preface
    1. Acknowledgments
  6. Chapter 1. Sets and Proofs
    1. Abstract
    2. 1.1 Sets, Elements, and Subsets
    3. 1.2 Operations on Sets
    4. 1.3 Language of Logic
    5. 1.4 Techniques of Proof
    6. 1.5 Relations
    7. 1.6 Functions
    8. 1.7* Axioms of Set Theory
    9. Exercises
  7. Chapter 2. Numbers
    1. Abstract
    2. 2.1 System <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">N</span>
    3. 2.2 Systems <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">Z</span> and and <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">Q</span>
    4. 2.3 Least Upper Bound Property and <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">Q</span>
    5. 2.4 System <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">R</span>
    6. 2.5 Least Upper Bound Property and <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">R</span>
    7. 2.6* Systems <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">R&#175;</span>, , <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">C</span>, and , and <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">&#8727;R</span>
    8. 2.7 Cardinality
    9. Exercises
  8. Chapter 3. Convergence
    1. Abstract
    2. 3.1 Convergence of Numerical Sequences
    3. 3.2 Cauchy Criterion for Convergence
    4. 3.3 Ordered Field Structure and Convergence
    5. 3.4 Subsequences
    6. 3.5 Numerical Series
    7. 3.6 Some Series of Particular Interest
    8. 3.7 Absolute Convergence
    9. 3.8 Number <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">e</span>
    10. Exercises
  9. Chapter 4. Point Set Topology
    1. Abstract
    2. 4.1 Metric Spaces
    3. 4.2 Open and Closed Sets
    4. 4.3 Completeness
    5. 4.4 Separability
    6. 4.5 Total Boundedness
    7. 4.6 Compactness
    8. 4.7 Perfectness
    9. 4.8 Connectedness
    10. 4.9* Structure of Open and Closed Sets
    11. Exercises
  10. Chapter 5. Continuity
    1. Abstract
    2. 5.1 Definition and Examples
    3. 5.2 Continuity and Limits
    4. 5.3 Continuity and Compactness
    5. 5.4 Continuity and Connectedness
    6. 5.5 Continuity and Oscillation
    7. 5.6 Continuity of <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">Rk</span> -valued Functions -valued Functions
    8. Exercises
  11. Chapter 6. Space <em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">C</em>((<em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">E</em>,,<em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">E</em>&#8242;)′)
    1. Abstract
    2. 6.1 Uniform Continuity
    3. 6.2 Uniform Convergence
    4. 6.3 Completeness of <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">C(E,E&#8242;)</span>
    5. 6.4 Bernstein and Weierstrass Theorems
    6. 6.5* Stone and Weierstrass Theorems
    7. 6.6* Ascoli<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">-</span>Arzel&#224; TheoremArzelà Theorem
    8. Exercises
  12. Chapter 7. Differentiation
    1. Abstract
    2. 7.1 Derivative
    3. 7.2 Differentiation and Continuity
    4. 7.3 Rules of Differentiation
    5. 7.4 Mean-Value Theorems
    6. 7.5 Taylor’s Theorem
    7. 7.6* Differential Equations
    8. 7.7* Banach Spaces and the Space <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">C1(a,b)</span>
    9. 7.8 A View to Differentiation in <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="hiddenClass">Rk</span>
    10. Exercises
  13. Chapter 8. Bounded Variation
    1. Abstract
    2. 8.1 Monotone Functions
    3. 8.2 Cantor Function
    4. 8.3 Functions of Bounded Variation
    5. 8.4 Space <em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">BV</em>((<em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">a</em>, , <em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">b</em>))
    6. 8.5 Continuous Functions of Bounded Variation
    7. 8.6 Rectifiable Curves
    8. Exercises
  14. Chapter 9. Riemann Integration
    1. Abstract
    2. 9.1 Definition of the Riemann Integral
    3. 9.2 Existence of the Riemann Integral
    4. 9.3 Lebesgue Characterization
    5. 9.4 Properties of the Riemann Integral
    6. 9.5 Riemann Integral Depending on a Parameter
    7. 9.6 Improper Integrals
    8. Exercises
  15. Chapter 10. Generalizations of Riemann Integration
    1. Abstract
    2. 10.1 Riemann– Stieltjes Integral
    3. 10.2* Helly’s Theorems
    4. 10.3* Reisz Representation
    5. 10.4* Definition of the Kurzweil– Henstock Integral
    6. 10.5* Differentiation of the Kurzweil– Henstock Integral
    7. 10.6* Lebesgue Integral
    8. Exercises
  16. Chapter 11. Transcendental Functions
    1. Abstract
    2. 11.1 Logarithmic and Exponential Functions
    3. 11.2* Multiplicative Calculus
    4. 11.3 Power Series
    5. 11.4 Analytic Functions
    6. 11.5 Hyperbolic and Trigonometric Functions
    7. 11.6 Infinite Products
    8. 11.7* Improper Integrals Depending on a Parameter
    9. 11.8* Euler’s Integrals
    10. Exercises
  17. Chapter 12. Fourier Series and Integrals
    1. Abstract
    2. 12.1 Trigonometric Series
    3. 12.2 Riemann–Lebesgue Lemma
    4. 12.3 Dirichlet Kernels and Riemann’s Localization Lemma
    5. 12.4 Pointwise Convergence of Fourier Series
    6. 12.5* Fourier Series in Inner Product Spaces
    7. 12.6* Cesàro Summability and Fejér’s Theorem
    8. 12.7 Uniform Convergence of Fourier Series
    9. 12.8* Gibbs Phenomenon
    10. 12.9* Fourier Integrals
    11. Exercises
  18. Bibliography