实分析：长篇数学教材（长篇数学教材系列）第二版

English | 449 pages | Independently published; 2nd edition (July 15, 2019) | 1077254547 | PDF | 23 Mb

This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and includes interesting historical notes, periodic attempts at humor, and occasional diversions into other interesting areas of mathematics. The text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis. The author believes most textbooks are extremely overpriced and endeavors to help change this.Hints and solutions to select exercises can be found at LongFormMath.com.

This is the 2 + epsilon edition of this book. The second edition was published in July 2019. In January 2024, an epsilon of changes were made and the manuscript was updated, without officially creating a new edition.

英文|449页|独立出版；第二版（2019年7月15日）| 1077254547| PDF | 23 Mb 这本教科书是为学生设计的。本文不是典型的定义定理证明重复式，而是包含了更多的评论、动机和解释。这些证明并不简洁，旨在理解经济。此外，在数十个样张之前，会先进行“草稿工作”或样张草图，让学生了解全局，并解释他们将如何自己完成。例子往往会推动叙事，挑战读者的直觉。该文本还旨在使这些想法可见，并包含200多幅插图。写作很轻松，包括有趣的历史笔记、定期的幽默尝试，以及偶尔转向其他有趣的数学领域。本文涵盖了实数、基数、序列、级数、实数拓扑、连续性、微分、积分以及函数序列和级数。每一章都以练习结束，几乎都包括一些开放性问题。第一个附录包含一个结构reals，第二个是从分析中收集的额外的特殊和病理例子。作者认为，大多数教科书都定价过高，并努力帮助改变这一点。可以在LongFormMath.com上找到选择练习的提示和解决方案。 这是这本书的2+epsilon版本。第二版于2019年7月出版。2024年1月，进行了一系列修改，手稿也进行了更新，但没有正式创建新版本。

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