Introduction to Real Analysis

This classic textbook has been used successfully by instructors and students for nearly three decades. This timely new edition offers minimal yet notable changes while retaining all the elements, presentation, and accessible exposition of previous editions. A list of updates is found in the Preface to this edition.

This text is based on the author's experience in teaching graduate courses and the minimal requirements for successful graduate study. The text is understandable to the typical student enrolled in the course, taking into consideration the variations in abilities, background, and motivation. Chapters one through six have been written to be accessible to the average student,

w hile at the same time challenging the more talented student through the exercises.

Chapters seven through ten assume the students have achieved some level of expertise in the subject. In these chapters, the theorems, examples, and exercises require greater sophistication and mathematical maturity for full understanding.

In addition to the standard topics the text includes topics that are not always included in comparable texts.

• Chapter 6 contains a section on the Riemann-Stieltjes integral and a proof of Lebesgue's t heorem providing necessary and sufficient conditions for Riemann integrability.
• Chapter 7 also includes a section on square summable sequences and a brief introduction to normed linear spaces.
• C hapter 8 contains a proof of the Weierstrass approximation theorem using the method of

aapproximate identities.

• The inclusion of Fourier series in the text allows the student to gain some exposure to this important subject.
• The final chapter includes a detailed treatment of Lebesgue measure and the Lebesgue integral, using inner and outer measure.
• The exercises at the end of each section reinforce the concepts.
• Notes provide historical comments or discuss additional topics.

這本經典教科書已經成功地被教師和學生使用了近三十年。這個及時的新版本在保留了前幾版的所有元素、呈現方式和易於理解的解釋的同時，進行了一些微小但值得注意的改變。有關更新的列表可以在本版本的前言中找到。

這本教材基於作者在教授研究生課程以及成功完成研究生學習的最低要求的經驗。考慮到學生的能力、背景和動機的差異，這本教材對於修讀該課程的典型學生來說是易於理解的。第一章到第六章的內容寫得平易近人，同時通過練習題來挑戰更有才華的學生。

第七章到第十章假設學生在該主題上已經達到了一定的專業水平。在這些章節中，定理、例子和練習題需要更高的數學造詣和數學成熟度才能完全理解。

除了標準的主題外，本教材還包括一些在其他相似教材中不常見的主題。

第六章包含了一節關於黎曼-斯蒂爾捷積分以及黎曼可積條件的勒貝格定理的證明。

第七章還包括一節關於平方可加序列和對偶線性空間的簡要介紹。

第八章通過使用近似恆等式的方法證明了韋伊斯特拉斯逼近定理。

教材中包含傅立葉級數的內容，讓學生對這一重要主題有所了解。

最後一章詳細介紹了勒貝格測度和勒貝格積分，使用內部和外部測度。

每個章節結束時的練習題強化了概念的理解。

註釋提供了歷史評論或討論其他主題。

Manfred Stoll received his Ph.D. from Penn State University under the supervision of K.T. Hahn and supported by an NDEA Fellowship. He has spent his entire career at University of South Carolina, serving as Department Chair, and supervised 3 master's students and eight doctoral students. He has published over 47 refereed research articles and published three books. He has served as a Referee on over 22 journals, served on numerous Panel Sessions for the AMS and Review Panels for the Science Foundation of Ireland (2005-2008), and was the Program Officer in Mathematics and Physics for the Science Foundation of Ireland in Summer 2007. Since 1990 he has given 27 invited conference talks, including hour addresses and plenary talks.

Manfred Stoll在K.T. Hahn的指導下，並獲得NDEA Fellowship的支持，於賓夕法尼亞州立大學獲得博士學位。他的整個職業生涯都在南卡羅來納大學度過，曾擔任系主任，指導了3名碩士生和8名博士生。他發表了超過47篇經過同行評審的研究文章，並出版了三本書籍。他曾擔任22個期刊的審稿人，並參與了美國數學學會的多個專題討論會和愛爾蘭科學基金會（2005-2008）的評審小組，並在2007年夏季擔任愛爾蘭科學基金會的數學和物理項目主任。自1990年以來，他已經發表了27次受邀的會議演講，包括長時間演講和主題演講。