The Big Book of Real Analysis: From Numbers to Measures
Johar, Syafiq
- 出版商: Springer
- 出版日期: 2024-01-05
- 售價: $4,470
- 貴賓價: 9.5 折 $4,247
- 語言: 英文
- 頁數: 938
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 303130831X
- ISBN-13: 9783031308314
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相關主題
商品描述
This book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus.
Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the studyof measures and how one can construct the Lebesgue integral as an extension of the Riemann integral.
This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study.
商品描述(中文翻譯)
這本書提供了對實分析的介紹,這是數學研究中的基礎主題,也是必修的學科。它涉及無窮和極限的概念,這些是微積分發展的基石。
從一些基本的證明技巧和集合、函數的概念開始,本書從自然數嚴謹地構建了實數及其相關結構。在這個過程中,讀者將遇到無窮、極限、實數序列和實數級數的概念。然後,這些概念被形式化並作為獨立的對象進行討論。最後,它們被擴展到更一般的對象,如實值函數的極限、序列和級數。一旦建立了基本的工具,讀者將從基本原理進入對微積分(連續性、微分和黎曼積分)的經典研究。本書以介紹測度和如何構建勒貝格積分作為黎曼積分的擴展來結束。
這本教科書針對數學本科生。正如標題所示,它涵蓋了大量的材料,可以在大約三個學期內教授。許多備註和例子有助於激發和提供對抽象理論概念的直觀理解。此外,書中還包含600多個練習題,其中一些將引導讀者進一步研究更高級的主題,並適合獨立研究項目。由於本書是完全自包含的,也非常適合自學。
作者簡介
作者簡介(中文翻譯)
Syafiq Johar畢業於倫敦帝國學院,獲得純數學學士和碩士學位,之後在牛津大學完成了博士學位,研究了Ricci流的奇異形成。在牛津期間,他曾在聖愛德蒙廳和基督教堂擔任助教,這鍛煉了他對教授數學的熱情。在普林斯頓大學擔任Fleet Fellow一年後,他回到馬來西亞,在馬來西亞國立大學任職。