Analysis: I Analysis I
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Delhi
Hindustan Book Agency
[2022]
|
| Edition: | Fourth edition |
| Series: | Texts and readings in mathematics
37 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvi, 355 Seiten |
| ISBN: | 9788195196197 |
Staff View
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Record in the Search Index
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Contents Preface to the first edition Preface to subsequent editions 1 2 Introduction 1 1 2 The Peano axioms. Addition. Multiplication. Set theory 14 23 29 32 76 The integers. 76 The rationals. 83 Absolute value and exponentiation. 88 Gaps in the rational numbers. 92 The real numbers 5.1 5.2 5.3 12 Fundamentals. 32 Russell’s paradox (Optional).45 Functions. 48 Images and inverse images. 57 Cartesian products . 63 Cardinality of sets. 69 Integers and rationale 4.1 4.2 4.3 4.4 5 What is analysis?. Why do analysis?. Starting at the beginning: the natural numbers 3.1 3.2 3.3 3.4 3.5 3.6 4 xvii 1.1 1.2 2.1 2.2 2.3 3 xi 96 Cauchy
sequences. 98 Equivalent Cauchy sequences. 102 The construction of the real numbers. 105
Contents viii 5.4 5.5 5.6 Ordering the reals. 113 The least upper bound property. 119 Real exponentiation, part I. 124 6 Limits of sequences 129 6.1 Convergence and limit laws. 129 6.2 The Extended real number system. 136 6.3 Suprema and infima of sequences .140 6.4 Limsup, Liminf, and limit points . . . 142 6.5 Some standard limits. 151 6.6 Subsequences. 152 6.7 Real exponentiation, part II. 155 7 Series 159 7.1 Finite series. 159 7.2 Infinite series. 169 7.3 Sums of non-negative numbers. 174 7.4 Rearrangement of series. 178 7.5 The root and ratio tests. 182 8 Infinite sets 185 8.1 Countability. 185 8.2 Summation on infinite sets. 192 8.3 Uncountable
sets. 199 8.4 The axiom of choice. 202 8.5 Ordered sets. 206 9 Continuous functions on R 215 9.1 Subsets of the real line. 215 9.2 The algebra of real-valued functions. 222 9.3 Limiting values of functions . 224 9.4 Continuous functions. 231 9.5 Left and right limits. 235 9.6 The maximum principle. 238 9.7 The intermediate value theorem.242 9.8 Monotonic functions. 245 9.9 Uniform continuity . 247 9.10 Limits at infinity. 253
Contents ix 10 Differentiation of functions 255 10.1 Basic definitions.255 10.2 Local maxima, local minima, and derivatives. 261 10.3 Monotone functions and derivatives.264 10.4 Inverse functions and derivatives. 265 10.5 L’Höpital’s rule . 268 11 The 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 Riemann integral 270 Partitions. 271 Piecewise constant functions. 275 Upper and lower Riemann integrals. 279 Basic properties of the Riemann integral. 283 Riemann integrability of continuous functions . 288 Riemann integrability of monotonefunctions. 291 A non-Riemann integrable function. 293 The Riemann-Stieltjes integral. 295 The two fundamental theorems of calculus. 298 Consequences of the fundamental theorems. 303 A Appendix: the basics of mathematical logic 308 A.l Mathematical statements.309 A. 2 Implication. 315 A.3 The structure of proofs.
. 320 A.4 Variables and quantifiers. 323 A.5 Nested quantifiers. 327 A.6 Some examples of proofs and quantifiers . 330 A.7 Equality.332 В Appendix: the decimal system 335 B.l The decimal representation ofnatural numbers. 336 B.2 The decimal representation of real numbers. 339 Index 343 |
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| index_date | 2024-07-03T22:27:21Z |
| indexdate | 2024-11-22T17:48:53Z |
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| isbn | 9788195196197 |
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| spelling | Tao, Terence 1975- Verfasser (DE-588)132190370 aut Analysis I Analysis I Terence Tao ; Terence Tao Fourth edition New Delhi Hindustan Book Agency [2022] xvi, 355 Seiten txt rdacontent n rdamedia nc rdacarrier Texts and readings in mathematics 37 Texts and readings in mathematics ... (DE-604)BV022931971 1 Texts and readings in mathematics 37 (DE-604)BV011380811 37 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=034333548&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tao, Terence 1975- Analysis Texts and readings in mathematics |
| title | Analysis |
| title_auth | Analysis |
| title_exact_search | Analysis |
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| title_full | Analysis I Analysis I Terence Tao ; Terence Tao |
| title_fullStr | Analysis I Analysis I Terence Tao ; Terence Tao |
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| title_short | Analysis |
| title_sort | analysis analysis i |
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