Mathematical analysis:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
[2017]
|
| Series: | Foundations of applied mathematics
Volume 1 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xx, 689 Seiten Illustrationen, Diagramme |
| ISBN: | 9781611974898 |
Staff View
MARC
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| 100 | 1 | |a Humpherys, Jeffrey |4 aut | |
| 245 | 1 | 0 | |a Mathematical analysis |c Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans, Brigham Young University |
| 264 | 1 | |a Philadelphia |b Society for Industrial and Applied Mathematics |c [2017] | |
| 264 | 4 | |c © 2017 | |
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Record in the Search Index
| _version_ | 1804177743320973312 |
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| adam_text | Contents
List of Notation ix
Preface xiii
I Linear Analysis I 1
1 Abstract Vector Spaces 3
1.1 Vector Algebra............................................... 3
1.2 Spans and Linear Independence............................... 10
1.3 Products, Sums, and .Complements............................ 14
1.4 Dimension, Replacement, and Extension....................... 17
1.5 Quotient Spaces............................................. 21
Exercises.......................................................... 27
2 Linear Transformations and Matrices 31
2.1 Basics of Linear Transformations I.......................... 32
2.2 Basics of Linear Transformations II......................... 36
2.3 Rank, Nullity, and the First Isomorphism Theorem............ 40
2.4 Matrix Representations...................................... 46
2.5 Composition, Change of Basis, and Similarity ............... 51
2.6 Important Example: Bernstein Polynomials ................... 54
2.7 Linear Systems ............................................. 58
2.8 Determinants I.............................................. 65
2.9 Determinants II............................................. 70
Exercises.......................................................... 78
3 Inner Product Spaces 87
3.1 Introduction to Inner Products.............................. 88
3.2 Orthonormal Sets and Orthogonal Projections................. 94
3.3 Gram-Schmidt Orthonormalization............................. 99
3.4 QR with Householder Transformations....................... 105
3.5 Normed Linear Spaces .......................................110
3.6 Important Norm Inequalities.................................117
3.7 Adjoints....................................................120
3.8 Fundamental Subspaces of a Linear Transformation............123
3.9 Least Squares...............................................127
Exercises..........................................................131
V
Contents
4 Spectral Theory 139
4.1 Eigenvalues and Eigenvectors ................................140
4.2 Invariant Subspaces..........................................147
4.3 Diagonalization............................................ 150
4.4 Schur’s Lemma ...............................................155
4.5 The Singular Value Decomposition ............................159
4.6 Consequences of the SVD......................................165
Exercises..........................................................171
II Nonlinear Analysis I 177
5 Metric Space Topology 179
5.1 Metric Spaces and Continuous Functions.......................180
5.2 Continuous Functions and Limits..............................185
5.3 Closed Sets, Sequences, and Convergence......................190
5.4 Completeness and Uniform Continuity..........................195
5.5 Compactness..................................................203
5.6 Uniform Convergence and Banach Spaces........................210
5.7 The Continuous Linear Extension Theorem......................213
5.8 Topologically Equivalent Metrics.............................219
5.9 Topological Properties ......................................222
5.10 Banach-Valued Integration....................................227
Exercises..........................................................233
6 Differentiation 241
6.1 The Directional Derivative...................................241
6.2 The Frechet Derivative in Rn.................................246
6.3 The General Frechet Derivative...............................252
6.4 Properties of Derivatives....................................256
6.5 Mean Value Theorem and Fundamental Theorem of Calculus . . . 260
6.6 Taylor’s Theorem.............................................265
Exercises..........................................................272
7 Contraction Mappings and Applications 277
7.1 Contraction Mapping Principle................................278
7.2 Uniform Contraction Mapping Principle .......................281
7.3 Newton’s Method..............................................285
7.4 The Implicit and Inverse Function Theorems...................293
7.5 Conditioning............................................... 301
Exercises..........................................................310
III Nonlinear Analysis II 317
8 Integration I 319
8.1 Multivariable Integration.................................. 320
8.2 Overview of Daniell—Lebesgue Integration ....................326
8.3 Measure Zero and Measurability...............................331
Contents
vii
8.4 Monotone Convergence and Integration on
Unbounded Domains............................................335
8.5 Fatou’s Lemma and the Dominated Convergence Theorem . . . 340
8.6 Fubini’s Theorem and Leibniz’s Integral Rule .................344
8.7 Change of Variables...........................................349
Exercises...........................................................356
9 * Integration II 361
9.1 Every Normed Space Has a Unique Completion...........361
9.2 More about Measure Zero.....................................364
9.3 Lebesgue-Integrable Functions.................................367
9.4 Proof of Fubini’s Theorem.....................................372
9.5 Proof of the Change of Variables Theorem......................374
Exercises......................................................... 378
10 Calculus on Manifolds 381
10.1 Curves and Arclength........................................ 381
10.2 Line Integrals.............................................. 386
10.3 Parametrized Manifolds......................................389
10.4 * Integration on Manifolds....................................393
10.5 Green’s Theorem ..............................................396
Exercises......................................................... 403
11 Complex Analysis 407
11.1 Holomorphic Functions.........................................407
11.2 Properties and Examples.......................................411
11.3 Contour Integrals.............................................416
11.4 Cauchy’s Integral Formula.....................................424
11.5 Consequences of Cauchy’s Integral Formula.....................429
11.6 Power Series and Laurent Series...............................433
11.7 The Residue Theorem...........................................438
11.8 * The Argument Principle and Its Consequences.................445
Exercises...........................................................451
IV Linear Analysis II 457
12 Spectral Calculus 459
12.1 Projections...................................................460
12.2 Generalized Eigenvectors......................................465
12.3 The Resolvent.................................................470
12.4 Spectral Resolution...........................................475
12.5 Spectral Decomposition I......................................480
12.6 Spectral Decomposition II................................... 483
12.7 Spectral Mapping Theorem......................................489
12.8 The Perron-Frobenius Theorem..................................494
12.9 The Drazin Inverse..........................................500
12.10 * Jordan Canonical Form.......................................506
Exercises...........................................................511
viii
Contents
13 Iterative Methods 519
13.1 Methods for Linear Systems................................520
13.2 Minimal Polynomials and Krylov Subspaces..................526
13.3 The Arnoldi Iteration and GMRES Methods...................530
13.4 * Computing Eigenvalues I.................................538
13.5 * Computing Eigenvalues II................................543
Exercises........................................................548
14 Spectra and Pseudospectra 553
14.1 The Pseudospectrum........................................554
14.2 Asymptotic and Transient Behavior.........................561
14.3 * Proof of the Kreiss Matrix Theorem......................566
Exercises........................................................570
15 Rings and Polynomials 573
15.1 Definition and Examples...................................574
15.2 Euclidean Domains.........................................583
15.3 The Fundamental Theorem of Arithmetic.....................588
15.4 Homomorphisms.............................................592
15.5 Quotients and the First Isomorphism Theorem...............598
15.6 The Chinese Remainder Theorem.............................601
15.7 Polynomial Interpolation and Spectral Decomposition.......610
Exercises........................................................618
V Appendices 625
A Foundations of Abstract Mathematics 627
A.l Sets and Relations........................................627
A.2 Functions............................................. 635
A.3 Orderings.................................................643
A.4 Zorn’s Lemma, the Axiom of Choice, and Well Ordering .... 647
A. 5 Cardinality...............................................648
B The Complex Numbers and Other Fields 653
B. l Complex Numbers...........................................653
B. 2 Fields....................................................659
C Topics in Matrix Analysis 663
C. l Matrix Algebra............................................663
C.2 Block Matrices............................................665
C.3 Cross Products ...........................................667
D The Greek Alphabet 669
Bibliography 671
Index
679
|
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| id | DE-604.BV044436948 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:52:56Z |
| institution | BVB |
| isbn | 9781611974898 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029838223 |
| oclc_num | 1002237699 |
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| owner | DE-91G DE-BY-TUM DE-384 DE-739 DE-83 |
| owner_facet | DE-91G DE-BY-TUM DE-384 DE-739 DE-83 |
| physical | xx, 689 Seiten Illustrationen, Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | marc |
| series | Foundations of applied mathematics |
| series2 | Foundations of applied mathematics |
| spelling | Humpherys, Jeffrey aut Mathematical analysis Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans, Brigham Young University Philadelphia Society for Industrial and Applied Mathematics [2017] © 2017 xx, 689 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Foundations of applied mathematics Volume 1 Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Jarvis, Tyler Jamison 1966- (DE-588)173866883 aut Evans, Emily J. aut Foundations of applied mathematics Volume 1 (DE-604)BV044436939 1 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029838223&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Humpherys, Jeffrey Jarvis, Tyler Jamison 1966- Evans, Emily J. Mathematical analysis Foundations of applied mathematics Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 |
| title | Mathematical analysis |
| title_auth | Mathematical analysis |
| title_exact_search | Mathematical analysis |
| title_full | Mathematical analysis Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans, Brigham Young University |
| title_fullStr | Mathematical analysis Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans, Brigham Young University |
| title_full_unstemmed | Mathematical analysis Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans, Brigham Young University |
| title_short | Mathematical analysis |
| title_sort | mathematical analysis |
| topic | Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029838223&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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