Basic Real Analysis
One of the bedrocks of any mathematics education, the study of real analysis introduces students both to mathematical rigor and to the deep theorems and counterexamples that arise from such rigor: for instance, the construction of number systems, the Cantor Set, the Weierstrass nowhere differentiabl...
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| Format: | eBook |
| Language: | English |
| Published: |
Boston, MA
Birkhäuser
2003, 2003
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| Edition: | 1st ed. 2003 |
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| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 8.3 Uniform Convergence and Limit Theorems
- 8.4 Power Series
- 8.5 Elementary Transcendental Functions
- 8.6 Fourier Series
- 8.7 Problems
- 9 Normed and Function Spaces
- 9.1 Norms and Normed Spaces
- 9.2 Banach Spaces
- 9.3 Hilbert Spaces
- 9.4 Function Spaces
- 9.5 Problems
- 10 The Lebesgue Integral (F. Riesz’s Approach)
- 10.1 Improper Riemann Integrals
- 10.2 Step Functions and Their Integrals
- 10.3 Convergence Almost Everywhere
- 10.4 The Lebesgue Integral
- 10.5 Convergence Theorems
- 10.6 The Banach Space L1
- 10.7 Problems
- 11 Lebesgue Measure
- 11.1 Measurable Functions
- 11.2 Measurable Sets and Lebesgue Measure
- 11.3 Measurability (Lebesgue’s Definition)
- 11.4 The Theorems of Egorov, Lusin, and Steinhaus
- 11.5 Regularity of Lebesgue Measure
- 11.6 Lebesgue’s Outer and Inner Measures
- 11.7 The Hilbert Spaces L2(E, % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv % 39gaiyaacqWFfcVraaa!47BC!
- 1 Set Theory
- 1.1 Rings and Algebras of Sets
- 1.2 Relations and Functions
- 1.3 Basic Algebra, Counting, and Arithmetic
- 1.4 Infinite Direct Products, Axiom of Choice, and Cardinal Numbers
- 1.5 Problems
- 2 Sequences and Series of Real Numbers
- 2.1 Real Numbers
- 2.2 Sequences in ?
- 2.3 Infinite Series
- 2.4 Unordered Series and Summability
- 2.5 Problems
- 3 Limits of Functions
- 3.1 Bounded and Monotone Functions
- 3.2 Limits of Functions
- 3.3 Properties of Limits
- 3.4 One-sided Limits and Limits Involving Infinity
- 3.5 Indeterminate Forms, Equivalence, Landau’s Little “oh” and Big “Oh”
- 3.6 Problems
- 4 Topology of ? and Continuity
- 4.1 Compact and Connected Subsets of ?
- 4.2 The Cantor Set
- 4.3 Continuous Functions
- 4.4 One-sided Continuity, Discontinuity, and Monotonicity
- 4.5 Extreme Value and Intermediate Value Theorems
- 4.6 Uniform Continuity
- 4.7 Approximation by Step, Piecewise Linear, and Polynomial Functions
- 4.8 Problems
- 5 Metric Spaces
- 5.1 Metrics and Metric Spaces
- 5.2 Topology of a Metric Space
- 5.3 Limits, Cauchy Sequences, and Completeness
- 5.4 Continuity
- 5.5 Uniform Continuity and Continuous Extensions
- 5.6 Compact Metric Spaces
- 5.7 Connected Metric Spaces
- 5.8 Problems
- 6 The Derivative
- 6.1 Differentiability
- 6.2 Derivatives of Elementary Functions
- 6.3 The Differential Calculus
- 6.4 Mean Value Theorems
- 6.5 L’Hôpital’s Rule
- 6.6 Higher Derivatives and Taylor’s Formula
- 6.7 Convex Functions
- 6.8 Problems
- 7 The Riemann Integral
- 7.1 Tagged Partitions and Riemann Sums
- 7.2 Some Classes of Integrable Functions
- 7.3 Sets of Measure Zero and Lebesgue’s Integrability Criterion
- 7.4 Properties of the Riemann Integral
- 7.5 Fundamental Theorem of Calculus
- 7.6 Functions of BoundedVariation
- 7.7 Problems
- 8 Sequences and Series of Functions
- 8.1 Complex Numbers
- 8.2 Pointwise and Uniform Convergence