求推荐数学分析和高等代数英文教材？

如果你是自学，我推荐你的学习顺序是按照MIT的课程

每门课程里都有syllabus和assignments

syllabus里面有Prerequisites,告诉你学习这门课程之前需要的内容，包含课题号，可递归去找到预备课程。

assignments里面有题目，都是教授针对这门课程出的题，大部分课程的assignments会有答案。

没有的你可以给教授发邮件，一般他都会给

我的学习顺序是 18.01、18.02、18.03、18.06、18.100A、18.700（中间还有学习物理，8.01，8.02以及耶鲁的 PSY200和PSY201）

18.100A（数学分析）和18.700（线性代数）是没有视频的，要买对应的书籍

另外，18.100对应有三门课程（18.100A, 18.100B, 18.100C）

textbook对应关系是:

18.100A ==> Mattuck's Analysis

18.100B ==> Rudin's Analysis

18.100C ==> Rundin's Analysis

我自学选择的是18.100A，原因如下：

1. textbook内容和上课所教的内容很贴近，textbook就是18.100A授课老师 Mattuck写的。我可以按照每一个assignments的session去学习，学习完了做题目加深理解。我之前看了Rudin的目录，和MIT的18.100B的授课顺序，很多内容都不一样，要系统的学习很不方便。

2. 18.03的视频公开课授课老师是Mattuck，后继学习18.100A连贯性更强，之前课程里提到的存在性定理等在后续课程都有提到。

3. 18.100A的习题更多，可以加深理解。

18.100A不同点：

18.100A比其它两门课程少了一部分内容：point-set topology。如果要学后续课程，可自己单独学习这一章。

18.100A的书写不如后两门课程严谨。书写方面，我有参考Rudin的部分章节。

以下是Mattuck教授给的学习意见（正常来说，有问题发邮件给他们，他们都会回复）：

Each course in American universities has a number and a subject name.

18.100A is the number, and Introduction to Analysis its subject name. (\

(The number is necessary, since there is another version of the course

with the same name, but numbered 18.100B, which is harder, uses a

different book, and would not be so suitable for self-study.)

Work on the assignments one-by-one on the 18.100A OpenCourseWare website.

Use as book the 8th printing by CreateSpace, which costs $15 at Amazon;

if you use an earlier printing (only available as a used copy), you will

need to use the link to the list of corrections.

Use the questions after each section to test whether you understood what

you read; they have answers or hints at the end of the chapter.

The website does not have solutions to the assignments; if you find you

need them, I can send you a few at a time, if you pledge to use them only

for yourself and not distribute them to others.

You can use the practice exams on the website as real exams, timing

yourself; students at MIT can use the textbook at these exams, but not any

other material (the assignments and solutions, for instance).

In general, the courses beyond the ones you have already taken do not have

the same support as the more elementary ones; they suppose a student is

more serious and has more mathematical maturity, which you seem to have.

It probably would help to have a mathematical friend you could consult if

there is something you don't follow; however the book is sufficiently

clear I think, so re-reading a passage should be enough to understand it.

另外，Jerison有说明具体的差别；

100A and 100B are essentially the same class (100B has a bit more of what

is known as point-set topology). As it happens we also have another version

of 100 called 100C. These are all flavors of the same subject.

18.101 is a sequel to 100A or 100B or 100C.

There are two other classes 18.102 and 18.103 in the analysis sequence.

These are different from 18.101, but they only need 100B or 100A as prerequisite.

In other words, after 100A or 100B or 100C you can take any of 101, 102 or 103, in any

order. There is a bit of overlap between 102 and 103. There is essentially no overlap

between 101 and the other two.

Jerison后继内容：

We give all three because it depends on the student which one is better.

100C is completely out the question for you. The extra is additional time

reviewing writing style (writing proofs), which cannot be done long distance.

The material is matched to 100B (but there is another version matched to 100A, whose

letter I forget --- maybe they are going to be renamed 100P and 100Q).

Please don't worry about this. The main distinction is that 100B is based

on Rudin's textbook and 100A is based on Mattuck's textbook.

Mattuck针对具体数学问题的回复：

The statement being proved in this part of the proof is:

L - e < M for any e > 0 ==> L <= M .

Its contrapositive is

L > M ==> L - e >= M for at least one e > 0 (e.g., e = L - M).

The contrapositive is clearly true, which is equivalent to the original

statement being proved true.

I hope this makes the argument clear, which was presented somewhat informally in the book. Such writing is normal, so it is important to get used to it, by seeing how it would look if made precise. Few books would write out the contrapositive so explicitly.

我不是MIT的学生，但他们都会回复你邮件，只要你有具体的问题和明确的目的性。

都说Rudin的分析是holy bible。但是我认为是要选适合自己的，承前启后的学习才最重要。经典的教材可以作为辅助的学习书籍。一定要有一个系统的主线。