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Real analysis /
"Aimed at advanced undergraduates and beginning graduate students, Real Analysis offers a rigorous yet accessible course in the subject. Carothers, presupposing only a modest background in real analysis or advanced calculus, writes with an informal style and incorporates historical commentary a...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge [UK] ; New York :
Cambridge University Press,
2000.
|
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Part 1 Metric Spaces
- 1 Calculus Review 3
- Real Numbers 3
- Limits and Continuity 14
- 2 Countable and Uncountable Sets 18
- Equivalence and Cardinality 18
- Cantor Set 25
- Monotone Functions 31
- 3 Metrics and Norms 36
- Metric Spaces 37
- Normed Vector Spaces 39
- More Inequalities 43
- Limits in Metric Spaces 45
- 4 Open Sets and Closed Sets 51
- Open Sets 51
- Closed Sets 53
- Relative Metric 60
- 5 Continuity 63
- Continuous Functions 63
- Homeomorphisms 69
- Space of Continuous Functions 73
- 6 Connectedness 78
- Connected Sets 78
- 7 Completeness 89
- Totally Bounded Sets 89
- Complete Metric Spaces 92
- Fixed Points 97
- Completions 102
- 8 Compactness 108
- Compact Metric Spaces 108
- Uniform Continuity 114
- Equivalent Metrics 120
- 9 Category 128
- Discontinuous Functions 128
- Baire Category Theorem 131
- Part 2 Function Spaces
- 10 Sequences of Functions 139
- Historical Background 139
- Pointwise and Uniform Convergence 143
- Interchanging Limits 150
- Space of Bounded Functions 153
- 11 Space of Continuous Functions 162
- Weierstrass Theorem 162
- Trigonometric Polynomials 170
- Infinitely Differentiable Functions 176
- Equicontinuity 178
- Continuity and Category 183
- 12 Stone-Weierstrass Theorem 188
- Algebras and Lattices 188
- Stone-Weierstrass Theorem 194
- 13 Functions of Bounded Variation 202
- Functions of Bounded Variation 202
- Helly's First Theorem 210
- 14 Riemann-Stieltjes Integral 214
- Weights and Measures 214
- Riemann-Stieltjes Integral 215
- Space of Integrable Functions 221
- Integrators of Bounded Variation 225
- Riemann Integral 232
- Riesz Representation Theorem 234
- Other Definitions, Other Properties 239
- 15 Fourier Series 244
- Dirichlet's Formula 250
- Fejer's Theorem 254
- Complex Fourier Series 257
- Part 3 Lebesgue Measure and Integration
- 16 Lebesgue Measure 263
- Problem of Measure 263
- Lebesgue Outer Measure 268
- Riemann Integrability 274
- Measurable Sets 277
- Structure of Measurable Sets 283
- A Nonmeasurable Set 289
- Other Definitions 292
- 17 Measurable Functions 296
- Measurable Functions 296
- Extended Real-Valued Functions 302
- Sequences of Measurable Functions 304
- Approximation of Measurable Functions 306
- 18 Lebesgue Integral 312
- Simple Functions 312
- Nonnegative Functions 314
- General Case 322
- Lebesgue's Dominated Convergence Theorem 328
- Approximation of Integrable Functions 333
- 19 Additional Topics 337
- Convergence in Measure 337
- L[subscript p] Spaces 342
- Approximation of L[subscript p] Functions 350
- More on Fourier Series 352
- 20 Differentiation 359
- Lebesgue's Differentiation Theorem 359
- Absolute Continuity 370.