The elements of real analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
1976
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| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 480 S. Ill., graph. Darst. |
| ISBN: | 047105464X |
Staff View
MARC
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| 245 | 1 | 0 | |a The elements of real analysis |c Robert G. Bartle |
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Record in the Search Index
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| adam_text | Titel: The elements of real analysis
Autor: Bartle, Robert G
Jahr: 1976
Chapter Summaries Introduction: A Glimpse at Set Theory 1 1. The Algebra of Sets, 1 Equality of sets, intersection, union, Cartesian product 2. Functions, 11 Tabular representation, transformations, restrictions and extensions, composition, injective and inverse functions, surjective and bijective functions, direct and inverse images 3. Finite and Infinite Sets, 22 Finite, countable, and uncountable sets, the uncountability of R and 1 I. The Real Numbers 27 4. The Algebraic Properties of R, 28 The field properties of R, irrationality of V2 5. The Order Properties of R, 32 Order properties, absolute value 6. The Completeness Property of R, 37 Suprema and infima, Archimedean Property, the existence of /2 7. Cuts, Intervals, and the Cantor Set, 45 The Cut Property, cells and intervals, Nested Property, the Cantor set, models for R II. The Topology of Cartesian Spaces 52 8. Vector and Cartesian Spaces, 52 Vector spaces, inner product spaces, normed spaces, the Schwarz Inequality, the Cartesian space R p 9. Open and Closed Sets, 62 Open sets, closed sets, neighborhoods 10. The Nested Cells and Bolzano-Weierstrass Theorem, 68 Nested Cells Theorem, cluster points, Bolzano-Weierstrass Theorem 11. The Heine-Borel Theorem, 72 Compactness, the Heine-Borel Theorem, Cantor Intersection Theorem, Lebesgue Covering Theorem
CHAPTER SUMMARIES 12. Connected Sets, 80 The connectedness of intervals in R, polygonally connected open sets are connected, connected sets in R are intervals 13. The Complex Number System, 86 Definition and elementary properties III. Convergence 90 14. Introduction to Sequences, 90 Convergence, uniqueness of the limit, examples 15. Subsequences and Combinations, 98 Subsequences, algebraic combinations of sequences 16. Two Criteria for Convergence, 104 Monotole Convergence Theorem, Bolzano-Weierstrass Theorem, Cauchy sequences, the Cauchy Criterion 17. Sequences of Functions, 113 Convergence, uniform convergence, the uniform norm, Cauchy Criterion for Uniform Convergence 18. The Limit Superior, 123 The limit superior and inferior of a sequence in R, unbounded sequences, infinite limits 19. Some Extensions, 128 Order of magnitude, Cesåro summation, double sequences, iterated limits IV. Continuous Functions 136 20. Local Properties of Continuous Functions, 136 Continuity at a point and on a set, the Discontinuity Criterion, combinations of functions 21. Linear Functions, 147 Linear functions, matrix representation, the norm 22. Global Properties of Continuous Functions, 150 Global Continuity Theorem, Preservation of Compactness, Preservation of Connectedness, Continuity of the Inverse Function Theorem, bounded continuous functions 23. Uniform Continuity and Fixed Points, 158 Uniform continuity, Lipschitz condition, Fixed Point Theorem for Contractions, Brouwer Fixed Point Theorem 24. Sequences of Continuous Functions, 165 Interchange of limit and continuity, approximation by step and piecewise linear functions, Bernstein polynomials, the Bernstein and Weierstrass Approximation Theorems
CHAPTER SUMMARIES XIII 25. Limits of Functions, 174 Deleted and non-deleted limits, the deleted and non-deleted limit inferior, semi-continuity 26. Some Further Results, 182 The Stone and Stone-Weierstrass Approximation Theorems, Polynomial Approximation Theorem, Tietze’s Extension Theorem, equicontinuity, Arzela-Ascoli Theorem V. Functions of One Variable 193 27. The Mean Value Theorem, 193 The derivative, Interior Maximum Theorem, Rolle’s Theorem, Mean Value Theorem 28. Further Applications of the Mean Value Theorem, 201 Applications, L’Hospital’s Rules, interchange of limit and derivative, Taylor’s Theorem 29. The Riemann-Stieltjes Integral, 212 Riemann-Stieltjes sums and the integral, Cauchy Criterion for Integrability, properties of the integral, integration by parts, modification of the integral 30. Existence of the Integral, 227 Riemann Criterion for Integrability, the integrability of continuous functions, Mean Value Theorems, Differentiation Theorem, Fundamental Theorem of Integral Calculus, Change of Variable Theorem 31. Further Properties of the Integral, 240 Interchange of limit and integral, Bounded Convergence Theorem, Monotone Convergence Theorem, integral form of the remainder, integrals depending on a parameter, Leibniz’s formula, Interchange Theorem, Riesz Representation Theorem 32. Improper and Infinite Integrals, 257 Improper integrals of unbounded functions, infinite integrals, Cauchy Criterion, Comparison Test, Limit Comparison Test, Dirichlet’s Test, absolute convergence 33. Uniform Convergence and Infinite Integrals, 267 Cauchy Criterion for uniform convergence, Weierstrass M-Test, Dirichlet’s Test, infinite integrals depending on a parameter, Dominated Convergence Theorem, iterated infinite integrals VI. Infinite Series 286 34. Convergence of Infinite Series, 286 Convergence of series, Cauchy Criterion, absolute convergence, Rearrangement Theorem
xiv CHAPTER SUMMARIES 35. Tests for Absolute Convergence, 294 Comparison Test, Limit Comparison Test, Root Test, Ratio Test, Raabe’s Test, Integral Test 36. Further Results for Series, 305 Abel’s Lemma, Dirichlet’s Test, Abel’s Test, Alternating Series Test, double series, Cauchy multiplication 37. Series of Functions, 315 Absolute and uniform convergence, Cauchy Criterion, Weierstrass M-Test, Dirichlet’s Test, Abel’s Test, power series, Cauchy-Hadamard Theorem, Differentiation Theorem, Uniqueness Theorem, Multiplication Theorem, Bernstein’s Theorem, Abel’s Theorem, Tauber’s Theorem 38. Fourier Series, 330 Bessel’s Inequality, Riemann-Lebesgue Lemma, Pointwise Convergence Theorem, Uniform Convergence Theorem, Norm Convergence Theorem, Parseval’s Equality, Fejer’s Theorem, Weierstrass Approximation Theorem VII. Differentiation in R p 346 39. The Derivative in R p , 347 Partial derivatives, directional derivatives, the derivative of ƒ : R p —* R q , the Jacobian 40. The Chain Rule and Mean Value Theorems, 360 Chain Rule, Mean Value Theorem, interchange of the order of differentiation, higher derivatives, Taylor’s Theorem 41. Mapping Theorems and Implicit Functions, 375 Class C , Approximation Lemma, Injective Mapping Theorem, Surjective Mapping Theorem, Open Mapping Theorem, Inversion Theorem, Implicit Function Theorem, Parametrization Theorem, Rank Theorem 42. Extremum Problems, 397 Relative extrema, Second Derivative Test, extremum problems with constraints, Lagrange’s Theorem, inequality constraints VIII. Integration in R p 412 43. The Integral in R p , 412 Content zero, Riemann sums and the integral, Cauchy Criterion, properties of the integral, Integrability Theorem 44. Content and the Integral, 422 Sets with content, characterization of the content function, further properties of the integral, Mean Value Theorem, iterated integrals
CHAPTER SUMMARIES 45. Transformation of Sets and Integrals, 437 Images of sets with content under C maps, transformations by linear maps, transformations by non-linear maps, the Jacobian Theorem, Change of Variables Theorem, polar and spherical coordinates, strong form of the Change of Variables Theorem References 456 Hints for Selected Exercises 458 Index 475
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| author | Bartle, Robert G. 1927-2003 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.8 |
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| dewey-sort | 3515 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| isbn | 047105464X |
| language | English |
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| physical | XV, 480 S. Ill., graph. Darst. |
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| spelling | Bartle, Robert G. 1927-2003 Verfasser (DE-588)13732877X aut The elements of real analysis Robert G. Bartle 2. ed. New York [u.a.] Wiley 1976 XV, 480 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Analise Real larpcal Analyse mathématique Analyse mathématique ram Mathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf Reelle Analysis (DE-588)4627581-2 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Analysis (DE-588)4001865-9 s DE-604 Reelle Analysis (DE-588)4627581-2 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001289057&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bartle, Robert G. 1927-2003 The elements of real analysis Analise Real larpcal Analyse mathématique Analyse mathématique ram Mathematical analysis Analysis (DE-588)4001865-9 gnd Reelle Analysis (DE-588)4627581-2 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4627581-2 (DE-588)4151278-9 |
| title | The elements of real analysis |
| title_auth | The elements of real analysis |
| title_exact_search | The elements of real analysis |
| title_full | The elements of real analysis Robert G. Bartle |
| title_fullStr | The elements of real analysis Robert G. Bartle |
| title_full_unstemmed | The elements of real analysis Robert G. Bartle |
| title_short | The elements of real analysis |
| title_sort | the elements of real analysis |
| topic | Analise Real larpcal Analyse mathématique Analyse mathématique ram Mathematical analysis Analysis (DE-588)4001865-9 gnd Reelle Analysis (DE-588)4627581-2 gnd |
| topic_facet | Analise Real Analyse mathématique Mathematical analysis Analysis Reelle Analysis Einführung |
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