The Real Numbers : An Introduction to Set Theory and Analysis / by John Stillwell
| データ種別 | 電子書籍 |
|---|---|
| 出版者 | Cham : Springer International Publishing : Imprint: Springer |
| 出版年 | 2013 |
書誌詳細を非表示
| 書誌ID | OB01027470 |
|---|---|
| 本文言語 | 英語 |
| 一般注記 | The Fundamental Questions -- From Discrete to Continuous -- Infinite Sets -- Functions and Limits -- Open Sets and Continuity -- Ordinals -- The Axiom of Choice -- Borel Sets -- Measure Theory -- Reflections -- Bibliography -- Index. License restrictions may limit access Summary: While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions |
| 著者標目 | *Stillwell, John SpringerLink (Online service) |
| 統一書名標目 | Undergraduate Texts in Mathematics, |
| 件 名 | LCSH:Mathematics LCSH:Functions of real variables LCSH:History LCSH:Mathematical logic FREE:Mathematics FREE:Real Functions FREE:Mathematical Logic and Foundations FREE:History of Mathematical Sciences |
| 分 類 | LCC:QA331.5 DC23:515.8 |
| 巻冊次 | ISBN:9783319015774 RefWorks出力（各巻） print ; ISBN:9783319015767 RefWorks出力（各巻） |
| 資料種別 | 機械可読データファイル |
目次・あらすじ
類似資料
この資料の利用統計
このページへのアクセス回数：2回
※2020年8月16日以降