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Mathematical Analysis

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Mathematical Analysis

This text carefully leads the student through the basic topics of Real Analysis. Topics include metric spaces, open and closed sets, convergent sequences, function limits and continuity, compact sets, sequences and series of functions, power series, differentiation and integration, Taylor's theorem, total variation, rectifiable arcs, and sufficient conditions of integrability.

AnalysisMathematics
Author
Elias Zakon
Publisher
University of Windsor
Source
LibreTexts
License
CC BY
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 174 sections
  1. 01: Set Theory
  2. 1.01: Sets and Operations on Sets. Quantifiers
  3. 1.1.E: Problems in Set Theory (Exercises)
  4. 1.02: Relations. Mappings
  5. 1.2.E: Problems on Relations and Mappings (Exercises)
  6. 1.03: Sequences
  7. 1.04: Some Theorems on Countable Sets
  8. 1.4.E: Problems on Countable and Uncountable Sets (Exercises)
  9. 02: Real Numbers and Fields
  10. 2.01: Axioms and Basic Definitions
  11. 2.02: Natural Numbers. Induction
  12. 2.2.E: Problems on Natural Numbers and Induction (Exercises)
  13. 2.03: Integers and Rationals
  14. 2.04: Upper and Lower Bounds. Completeness
  15. 2.4.E: Problems on Upper and Lower Bounds (Exercises)
  16. 2.05: Some Consequences of the Completeness Axiom
  17. 2.06: Powers with Arbitrary Real Exponents. Irrationals
  18. 2.6.E: Problems on Roots Powers and Irrationals (Exercises)
  19. 2.07: The Infinities. Upper and Lower Limits of Sequences
  20. 2.7.E: Problems on Upper and Lower Limits of Sequences in (E) (Exercises)
  21. 03: Vector Spaces and Metric Spaces
  22. 3.01: The Euclidean n-Space E
  23. 3.1.E: Problems on Vectors in (En) (Exercises)
  24. 3.02: Lines and Planes in E
  25. 3.2.E: Problems on Lines and Planes in (En) (Exercises)
  26. 3.03: Intervals in E
  27. 3.3.E: Problems on Intervals in E (Exercises)
  28. 3.04: Complex Numbers
  29. 3.4.E: Problems on Complex Numbers (Exercises)
  30. 3.05: Vector Spaces. The Space C. Euclidean Spaces
  31. 3.5.E: Problems on Linear Spaces (Exercises)
  32. 3.06: Normed Linear Spaces
  33. 3.6.E: Problems on Normed Linear Spaces (Exercises)
  34. 3.07: Metric Spaces
  35. 3.7.E: Problems on Metric Spaces (Exercises)
  36. 3.08: Open and Closed Sets. Neighborhoods
  37. 3.8.E: Problems on Neighborhoods Open and Closed Sets (Exercises)
  38. 3.09: Bounded Sets. Diameters
  39. 3.9.E: Problems on Boundedness and Diameters (Exercises)
  40. 3.10: Cluster Points. Convergent Sequences
  41. 3.10.E: Problems on Cluster Points and Convergence (Exercises)
  42. 3.11: Operations on Convergent Sequences
  43. 3.11.E: Problems on Limits of Sequences (Exercises)
  44. 3.12: More on Cluster Points and Closed Sets. Density
  45. 3.12.E: Problems on Cluster Points Closed Sets and Density
  46. 3.13: Cauchy Sequences. Completeness
  47. 3.13.E: Problems on Cauchy Sequences
  48. 04: Function Limits and Continuity
  49. 4.01: Basic Definitions
  50. 4.1.E: Problems on Limits and Continuity
  51. 4.02: Some General Theorems on Limits and Continuity
  52. 4.2.E: More Problems on Limits and Continuity
  53. 4.03: Operations on Limits. Rational Functions
  54. 4.3.E: Problems on Continuity of Vector-Valued Functions
  55. 4.04: Infinite Limits. Operations in E
  56. 4.4.E: Problems on Limits and Operations in (E)
  57. 4.05: Monotone Function
  58. 4.5.E: Problems on Monotone Functions
  59. 4.06: Compact Sets
  60. 4.6.E: Problems on Compact Sets
  61. 4.07: More on Compactness
  62. 4.08: Continuity on Compact Sets. Uniform Continuity
  63. 4.8.E: Problems on Uniform Continuity Continuity on Compact Sets
  64. 4.09: The Intermediate Value Property
  65. 4.9.E: Problems on the Darboux Property and Related Topics
  66. 4.10: Arcs and Curves. Connected Sets
  67. 4.10.E: Problems on Arcs Curves and Connected Sets
  68. 4.11: Product Spaces. Double and Iterated Limits
  69. 4.11.E: Problems on Double Limits and Product Spaces
  70. 4.12: Sequences and Series of Functions
  71. 4.12.E: Problems on Sequences and Series of Functions
  72. 4.13: Absolutely Convergent Series. Power Series
  73. 4.13.E: More Problems on Series of Functions
  74. 05: Differentiation and Antidifferentiation
  75. 5.01: Derivatives of Functions of One Real Variable
  76. 5.1.E: Problems on Derived Functions in One Variable
  77. 5.02: Derivatives of Extended-Real Functions
  78. 5.2.E: Problems on Derivatives of Extended-Real Functions
  79. 5.03: L'Hopital's Rule
  80. 5.3.E: Problems on (Lprime) Hopital's Rule
  81. 5.04: Complex and Vector-Valued Functions on (E1)
  82. 5.4.E: Problems on Complex and Vector-Valued Functions on (E1)
  83. 5.05: Antiderivatives (Primitives Integrals)
  84. 5.5.E: Problems on Antiderivatives
  85. 5.06: Differentials. Taylors Theorem and Taylors Series
  86. 5.6.E: Problems on Tayior's Theorem
  87. 5.07: The Total Variation (Length) of a Function f - E1 E
  88. 5.7.E: Problems on Total Variation and Graph Length
  89. 5.08: Rectifiable Arcs. Absolute Continuity
  90. 5.8.E: Problems on Absolute Continuity and Rectifiable Arcs
  91. 5.09: Convergence Theorems in Differentiation and Integration
  92. 5.9.E: Problems on Convergence in Differentiation and Integration
  93. 5.10: Sufficient Condition of Integrability. Regulated Functions
  94. 5.10.E: Problems on Regulated Functions
  95. 5.11: Integral Definitions of Some Functions
  96. 5.11.E: Problems on Exponential and Trigonometric Functions
  97. 06: Differentiation on E and Other Normed Linear Spaces
  98. 6.01: Directional and Partial Derivatives
  99. 6.1.E: Problems on Directional and Partial Derivatives
  100. 6.02: Linear Maps and Functionals. Matrices
  101. 6.2.E: Problems on Linear Maps and Matrices
  102. 6.03: Differentiable Functions
  103. 6.3.E: Problems on Differentiable Functions
  104. 6.04: The Chain Rule. The Cauchy Invariant Rule
  105. 6.4.E: Further Problems on Differentiable Functions
  106. 6.05: Repeated Differentiation. Taylors Theorem
  107. 6.5.E: Problems on Repeated Differentiation and Taylor Expansions
  108. 6.06: Determinants. Jacobians. Bijective Linear Operators
  109. 6.6.E: Problems on Bijective Linear Maps and Jacobians
  110. 6.07: Inverse and Implicit Functions. Open and Closed Maps
  111. 6.7.E: Problems on Inverse and Implicit Functions Open and Closed Maps
  112. 6.08: Baire Categories. More on Linear Maps
  113. 6.8.E: Problems on Baire Categories and Linear Maps
  114. 6.09: Local Extrema. Maxima and Minima
  115. 6.9.E: Problems on Maxima and Minima
  116. 6.10: More on Implicit Differentiation. Conditional Extrema
  117. 6.10.E: Further Problems on Maxima and Minima
  118. 07: Volume and Measure
  119. 7.01: More on Intervals in E. Semirings of Sets
  120. 7.1.E: Problems on Intervals and Semirings
  121. 7.02: (mathcalC sigma)-Sets. Countable Additivity. Permutable Series
  122. 7.2.E: Problems on (mathcalC sigma) -Sets (sigma) -Additivity and Permutable Series
  123. 7.03: More on Set Families
  124. 7.3.E: Problems on Set Families
  125. 7.04: Set Functions. Additivity. Continuity
  126. 7.4.E: Problems on Set Functions
  127. 7.05: Nonnegative Set Functions. Premeasures. Outer Measures
  128. 7.5.E: Problems on Premeasures and Related Topics
  129. 7.06: Measure Spaces. More on Outer Measures
  130. 7.6.E: Problems on Measures and Outer Measures
  131. 7.07: Topologies. Borel Sets. Borel Measures
  132. 7.7.E: Problems on Topologies Borel Sets and Regular Measures
  133. 7.08: Lebesgue Measure
  134. 7.8.E: Problems on Lebesgue Measure
  135. 7.09: LebesgueStieltjes Measures
  136. 7.9.E: Problems on Lebesgue-Stieltjes Measures
  137. 7.10: Generalized Measures. Absolute Continuity
  138. 7.10.E: Problems on Generalized Measures
  139. 7.11: Differentiation of Set Functions
  140. 7.11.E: Problems on Vitali Coverings
  141. 08: Measurable Functions and Integration
  142. 8.01: Elementary and Measurable Functions
  143. 8.1.E: Problems on Measurable and Elementary Functions in ((S mathcalM))
  144. 8.02: Measurability of Extended-Real Functions
  145. 8.2.E: Problems on Measurable Functions in ((S mathcalM m))
  146. 8.03: Measurable Functions in ((S mathcalM m))
  147. 8.3.E: Problems on Measurable Functions in ((S mathcalM m))
  148. 8.04: Integration of Elementary Functions
  149. 8.4.E: Problems on Integration of Elementary Functions
  150. 8.05: Integration of Extended-Real Functions
  151. 8.5.E: Problems on Integration of Extended-Real Functions
  152. 8.06: Integrable Functions. Convergence Theorems
  153. 8.6.E: Problems on Integrability and Convergence Theorems
  154. 8.07: Integration of Complex and Vector-Valued Functions
  155. 8.7.E: Problems on Integration of Complex and Vector-Valued Functions
  156. 8.08: Product Measures. Iterated Integrals
  157. 8.8.E: Problems on Product Measures and Fubini Theorems
  158. 8.09: Riemann Integration. Stieltjes Integrals
  159. 8.9.E: Problems on Riemann and Stieltjes Integrals
  160. 8.10: Integration in Generalized Measure Spaces
  161. 8.10.E: Problems on Generalized Integration
  162. 8.11: The RadonNikodym Theorem. Lebesgue Decomposition
  163. 8.11.E: Problems on Radon-Nikodym Derivatives and Lebesgue Decomposition
  164. 8.12: Integration and Differentiation
  165. 8.12.E: Problems on Differentiation and Related Topics
  166. 09: Calculus Using Lebesgue Theory
  167. 9.01: L-Integrals and Antiderivatives
  168. 9.1.E: Problems on L-Integrals and Antiderivatives
  169. 9.02: More on L-Integrals and Absolute Continuity
  170. 9.2.E: Problems on L-Integrals and Absolute Continuity
  171. 9.03: Improper (Cauchy) Integrals
  172. 9.3.E: Problems on Cauchy Integrals
  173. 9.04: Convergence of Parametrized Integrals and Functions
  174. 9.4.E: Problems on Uniform Convergence of Functions and C-Integrals