数学分析入门
A First Course in Mathematical Analysis
David A. Brannan · 初版 2006 年
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数学分析（Mathematical Analysis，通常被称为高等微积分 Advanced Calculus）普遍被学生认为是数学系难度最大的课程之一。本书采用所谓基于序列的方法来讲解连续性、可微性与可积性，从而降低该学科的理解门槛。标准微积分课程中往往一笔带过的论题，在此都会得到深入透彻的剖析。例如，一个“连续”函数究竟指的是什么？又该如何给“积分”下一个严谨周密的定义？后一个问题往往是微积分课程中的难点之一——并且相当难以
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……Stephen Abbot 著《理解分析》（Understanding Analysis）；David Stirling 著《数学分析与证明》（Mathematical Analysis and Proof）；Burn 著《数与函数：走进分析》（Numbers and Functions: Steps into Analysis）；Steven Lay 著《分析学：证明方法导论》（Analysis: With an Introduction to Proof）；David Brannan 著《数学分析基础教程》（A First Course in Mathematical Analysis）。
…Understanding Analysis by Stephen Abbot. Mathematical Analysis and Proof by David Stirling. Numbers and Functions: Steps into Analysis by Burn. Analysis: With an Introduction to Proof by Steven Lay. A First Course in Mathematical Analysis by David Brannan.
……风格虽然偏老，但还不至于难懂到跟不上（这是个人偏好，但我发现自己读这种风格的教材效果好得多）：Davenport 的《高等算术》（The Higher Arithmetic）；Burkill 的《数学分析基础教程》（A First Course in Mathematical Analysis，以及 Burkill 和 Burkill 合著的《数学分析进阶教程》）；Hardy 和 Wright 的《数论导引》（An introduction to the theory of numbers）。
…in style but not to the point of being too difficult to follow (it's personal preference but I found I got on much better with this style of textbook): Davenport - The Higher Arithmetic Burkill - A First Course in Mathematical Analysis (and also A Second Course in Mathematical Analysis by Burkill and Burkill) Hardy and Wright - An introduction to the theory of numbers
……Abbott 写的导论，以及关于连续性、微分和积分的习题。我觉得对于一本本科教材来说，这是个很有意思的教学设计选择，特别是像普林斯顿的 MATH215 那样把它作为数学分析的第一门入门课教材时（我确实觉得这是一本极棒的书）。
…Abbott to introduction and exercises on continuity, differentiation, and integration. I thought that was an interesting pedagogical choice for an undergraduate book, especially when it is used for the very first course in mathematical analysis as in Princeton’s MATH215 (I do think it is a really nice book).
……需求，因为 Rudin 讨论的是广义欧几里得空间，而不单单是实数集。其他有用的书还包括 Bryant 的《又一本分析学导论》（Yet Another Introduction to Analysis）、Lara Alcock 的《如何思考分析学》（How to Think About Analysis）、Brannan 的《数学分析基础教程》（A First Course in Mathematical Analysis）、Jay Cummings 的《实分析：长篇数学教材》（Real Analysis: A Long-Form Mathematics Textbook）等。这些书没有一本是完美的。比如除了 Cummings 那本之外，其他都没包含必要的拓扑……
…needs because Rudin considers Euclidean Spaces proper, not just the reals. Other helpful books include "Yet Another Introduction to Analysis" by Bryant, "How to Think About Analysis" by Lara Alcock, "A First Course in Mathematical Analysis" by Brannan, "Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings" etc. None of these books are perfect. For example, none of them (but that of Cummings) include the necessary topological…
对于实分析，无论如何我都建议你入一本“baby Rudin”，但如果是为了自学实分析，我更推荐像 Burkill 的《数学分析基础教程》（A First Course in Mathematical Analysis）这样的书。其他靠谱的选择包括 Spivak 的《微积分》（Calculus，这实际上是一本实分析书，难度介于典型的大一微积分教材和“baby Rudin”之间），以及《真实数学……
For real analysis, I'd recommend that you pick up a copy of "baby Rudin" regardless, but for learning real analysis, I'd recommend a book like "A First Course in Mathematical Analysis" by Burkill. Other viable options include "Calculus" by Spivak (it's really a book on real analysis at a level somewhere between a typical freshman calculus book and "baby Rudin"), "Real Mathematical…
……对于一个没怎么受过正规高等教育的人来说合理吗？如果不合理，我依然会把它当作业余爱好。我该如何弥补/填补我数学教育中的空白？——数学书单：目前正在读 Burkhill 的《数学分析基础教程》来学微积分（用来复习和拓展高中内容），在那之后打算读：我大学那本线性代数教材（当时没学好）、Spivak 的《流形上的微积分》（Calculus on Manifolds），以及《微分方程及其应用……
…someone with little formal education a reasonable one? If it's not, this will still be my hobby. How can I remediate/fill in the gaps of my math education? - Book list for math: currently reading "A first course in mathematical analysis" by Burkhill for calculus (to refresh and extend high school), and then after that: my college linear algebra book (done wrong), Spivak's "Calculus on Manifolds", and "Differential Equations with Applications…