实分析0-10

Terence Tao 的实分析课程已经完成了一多半,我到今天正好将他已经贴出来的帖子全部看完了,习题在脑子里都做了一遍,之还有一两个没有想通。到目前为止,收获不小,想到此做一个小结。这里是课程主页的链接

第零节是关于测度论的复习,基本上都是已知的结论,其中关于littlewood的三个学习实分析的原则还是蛮有意思的,我是头一次听说,不过这些思想在学习过程中依然感受到了。测度论中有很多很规范的证明方法,而且这些方法常常跟选择有关。我在评论中问了陶教授一个问题,已经困扰了我一段时间:测度论中有的问题会要求证明一个西格玛代数具有什么什么性质,这时候有一种很强大的方法,就是将满足这个性质的集合类直接设出来,然后再证明它确实是个西格玛代数。此种方法在运用的时候,常常会感到结论不知从什么地方出来的。证明的过程不具有启发性。陶教授回答了不少,上面的问题,正常的处理方法应该是从这个西格玛代数中任取一个元素,然后直接证明该元素是满足目标的性质的。但问题是,我们对这个任意的元素很难有个确切的描述。事实上,我们不得对“可数交”,“可数并”这样的操作做不可数次才行。而这本身其实正式选择公理强大行的体现。因为即使利用上面的方法构造出了“更清楚”的证明,其过程也一定会很复杂,很难看。

第零点五节是Carathéodory测度扩张定理的一个新的证明,而最终引出了一个Lebesgue测度的新的引入方式。我跟向开南教授聊起这个,他笑着说:“其实也不简单,这个定理听起来好像更快一些,但是需要的知识却更多了。”我很有同感。在一个乘积空间上的乘积测度,然后再转换成[0,1]区间上的测度。无论怎样,这是一个挺有趣的角度。

第一节是关于关于符号测度和Radon-Nikodym-Lebesgue定理。本节从正常测度开始讲起,事实上为符号测度做铺垫,为对符号测度有一个更清楚的了解而写的一个引子。我更看重陶教授在这方面做的事情。而且整个课程他一直是此种风格。本节主体内容跟常见的教材差不多,我更喜欢夹杂在证明过程中的一些解释。比如在证明Radon-Nikodym-Lebesgue定理的时候,正文中讲述整个过程中“贪婪”的思想,很有意思。结尾附上了一个选学的内容,即Lebesgue分解定理的有限形式。这个跟陶教授以前的一篇帖子有很大关系,我跟人很喜欢这个想法,我把最后的习题写了下来

第二节是全新的内容,我本人没有任何基础,讲的是Banach-Tarski悖论。该悖论是说,一个三维空间的单位球可以被拆成有限个部分,然后仅仅通过平移和旋转,就变成两个单位球。这够令人惊讶的了!但是存在性的证明,并没有具体给出如何拆的方法。而且一步步的看到最后,多少有点被绕糊涂了,没有整体的感觉,有兴趣的恐怕还的看看原始论文了。这篇帖子做了一个很不错的尝试,将所有跟选择公理有关的文字用红色表达。后来有些帖子涉及选择公理比较多的,也是这般处理的。此帖子中我还发现了一个很不多的博客,是由berkeley的几个博士生和博士们合写的。名字叫做Secret Blogging Seminar,链接在这里。本节中用到的关于将一个自由群嵌入到一个李群中去的方法,那个博客中有一个更清楚的帖子,做了更详细的阐述,我还没来得及细看。

第三节的内容是大家都很熟悉的L^P空间。开始两页作者站在一个很高的观点上谈了有关一般的数学对象的研究方法的内容,我很喜欢。但是初学者还是略去先不要看了。L^P空间中避免不了不等式。除了复习了几个以前见过的之外,我在学习中注意了不等式取等号时刻的意义。另外,伪-范数(pseudometrics)以及近似范数(quasi-norms)的概念也挺有意思的。

第四节讲了一个表示定理,用以说明为什么可以只研究具体的西格玛代数。这一节还是比较抽象的,表示定理的建立一般都是为了看清楚结构,分类,从而把问题简化。这一节也不例外。

第五节更是老朋友了,Hilbert空间。本节中除了一般书籍中常见的知识之外,有一个Hanner不等式(习题6)我从来没有见过,该不等式加强了平行四边形法则。另一个值得注意的地方是,Riesz表示定理竟然可以跟Lebesgue-Radon-Nikodym定理互推。在正交基的存在定理中,传说中的选择公理又来了。

第六节讲的是对偶关系和Hahn-Banach定理。该定理的重要程度没法再强调了,在数学论文中,可以不用写参考文献直接使用。没有人不知道的。而关于对偶关系,前边我在帖子中写过一个利用弱*拓扑和Alaoglu定理来证明StoneCech紧化的帖子,跟ultrafilter有很大关系,取自John.B.Conway的书。本节中几个重点的例子也蛮重要的。还有就是最后一部分中Hahn-Banach定理的各种加强版。在这方面我手头最好的教材就是上面说的这本书了:A course in functional analysis GTM96.

第七节整节讲选择公理,我看得真是爽死了。要知道,我在很长时间里都希望找一个这方面的材料好好的学一下。本节中包括的内容有:良序集,序数,Zorn引理,超限归纳以及他们和选择公理之间一些互推关系。众所周知,上面这些多半等价,本节短短几页把其中关系写的清清楚楚。

第八节复习点集拓扑。我还算熟悉,基本上把精力都放在了ultrafilter出现的地方。,ultrafilter是我学到的一个很有趣的工具,以前写过一个介绍性的帖子。还应该提到的是“网”的概念。它是数列的一个加强版。以前常常会遇到,这回算是有了一个全面的了解了。值得注意的是,一个空间是紧致的等价于任意一个网都有收敛子网。这是区别于列紧性质的。

第九节探讨Baire纲定理以及由此引发的几个大定理:一致有界原理,开映射定理,闭图像定理。把这些神奇的定理复习了一遍,加深了理解。要说Banach的工作,最著名的就就是这么几个定理,当真是刺激的数学好长时间的发展。Banach空间上的几何是非常奇特的,应该有很多人在做专门的研究。Tim Gowers在20几岁的时候就提出过有趣的Banach空间做反例,本节有提到。最后一个部分有一个非线性的应用。非线性偏微分方程我上学期花过一个多月下功夫,可是依然半途而废,好不遗憾,我认真的把这部分看完了,希望今后能够捡起来。

第十节是关于拓扑空间中的紧致性质的探讨,无疑是非常重要的。分成了三个部分,而三个部分中常常会出现的关于ultrafilter的证明,我是最喜欢的。第一个部分是紧致性和Hausdorff性的关系,显然,以前我是没有注意到的。在某种意义上,他们有着对立的关系。最后的Alexander子基定理十分有用。第二部分讲述了乘积空间的紧致性的定理,即Tychonoff定理。不可避免的,选择公理由出现了。最后还在习题中介绍了另外一种证明方法,用到ultrafilter的方法。另外,box拓扑的概念是这么自然,怎么以前没自己研究研究呢。第三个部分是关于Arzelá-Ascoli定理,这定理重要的无以复加了。

到目前为止,我学到了很多东西,更重要的是,这些帖子给了我一个全新的视角去看问题。很多以往知道或者体听过的理论,我在这是第一次结合到了一起看。名师的意义就在此吧。

听说Terry年仅7岁的时候就养成了读数学书,题目全做的习惯。想想自己曾经有过的错误想法,汗颜。从现在开始倒还不晚。

这里面的习题我都做了,还是那句话,欢迎有同学一起讨论。无论怎样,在跟陶教授学习的过程中,他为我打开了一扇窗,窗外的世界很广阔,也很吸引人。

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19 Comments on “实分析0-10”


  1. […] s Weblog placed an interesting blog post on å® […]

  2. PDEbeginner Says:

    Dear Xiaochuan,

    Have you done the second part of exercise 17? Prof. Tao has given me the answer, but I still didn’t understand it.

    If it is right, we can obtain (L^{\infty}([0,1]))^{*} \neq L^{1}([0,1]) because [0,1] include (0,1/2), (1/2,3/4), (3/4,7/8) … positive measure set.

    Could you help to try to find out my problem? Thanks a lot!

  3. liuxiaochuan Says:

    PDEbeginner:
    Exercise 17 of which note?

  4. PDEbeginner Says:

    oops, sorry, when I was issuing this question, nearly went to bed and not had a fresh brain. Note 6. Thanks a lot :)

  5. liuxiaochuan Says:

    Dear PDEbeginner,
    You are more than welcome to present any questions here. I am glad to discuss with you, especially questions from Professor Tao’s blog.

    As to this question, since we alredy have a apace containing a countably infinite sequence of disjoint sets of positive measure. Suppose these sets are E_1,E_2,\cdots. Then for any element \{a_n\}_1^\infty\in l^\infty(\Bbb N), we can define the funtions in L^\infty as \frac{1}{m(E_n)}I_{E_n} accordingly, taking values \frac{1}{m(E_n)} in E_n and vanishes otherwise. By this bijection, we actually get a closed subspace within L^\infty, which, by the first part of this exercise, lead to a contradiction.

    What I am interested is, how do you do the first two questions, I am not quite sure that I fully get the proof of them, at least not satisfied.

  6. PDEbeginner Says:

    Dear Xiaochuan,

    Thank you very much for your answer! I think there is some problem in your construction: if lim_{n \rightarrow \infty} m(E_n)=0, then the closure of span\{latex 1/m(E_n) 1_{E_n}\} is not a supspace of L^{\infty}.

    If your argument is true, then (L^{\infty}([0,1]))^{*} \neq L^{1}([0,1]) by taking (0,1/2), (1/2,3/4), (3/4,7/8) …. But it is well known that (L^{\infty}([0,1]))^{*} = L^{1}([0,1]).

    As for the first two questions: The second one is easy, by taking the dual of reflexive identity. The first one is a little tricky: Suppose Y \subset X, clearly Y^{*} \subset X^{*}. Take any functional l on Y^{*}, by H-B thm, there exists at least one extension of l, denoted by \bar{l}. Since X is reflexive, there exists some x \in X such that l(x)=\bar{l}. Decompose x=x_1+x_2 with x_1 \in Y and ||x_2||=\inf_{y_1 \in Y}||x-y_1|| (actually such decomposition is unique), then apply some easy argument to identify l(x_1)=l.

  7. liuxiaochuan Says:

    Dear PDEbeginner,

    My answer to the first question is quite similar to yours, though not so clear as yours.

    As for the last one, there was indeed some problems with my previous answer. But I don’t think that (L^{\infty}([0,1]))^{*} = L^{1}([0,1]) is correct. Let C([0,1]) be the subspace of all the continuous functions in L^\infty, and dirac functional \delta_x so that \delta_x(f)=f(x). So \delta_x is a continuous linear functional but don’t belong to L^1.

    I modify the proof as following. Suppose we alredy have a space containing a countably infinite sequence of disjoint sets of positive measure. Suppose these sets are E_1,E_2,\cdots. Then for any element \{a_n\}_1^\infty \in l^1(\Bbb N), we can define the funtion in L^1 as \sum\frac{a_n}{m(E_n)}I_{E_n} accordingly. By this bijection, we actually get a closed subspace within L^1, and hence a copy in L^\infty.

  8. PDEbeginner Says:

    Dear Xiaochuan,

    I am now reading the lecture note 8 of real analysis on Prof. Tao’s blog. I have some problem on the claim in Example 12, i.e. f is a right continuous function iff f is a continuous map from ({\bf R}, \mathcal{F}_r) to ({\bf R}, \mathcal{F}). I thought it over and over again, but still not found any hint to prove it. I was wondering if you could help to give an answer!

    Thanks a lot in advance anyway!

  9. liuxiaochuan Says:

    Dear PDEbeginner:
    For one direction, our purpose is to prove that for any x_0\in Bbb {R}, the map f(x) is right-continuous, provided f is continuous from ({\bf R}, \mathcal{F}_r) to ({\bf R}, \mathcal{F}). We rewrite this as ‘\forall \varepsilon >0, there exists some \delta>0 so that |f(x)-f(x_0)|<\varepsilon for all x\in [x_0,x_0+\delta) ‘,which is true since the inverse of an intervel (f(x_0)-\varepsilon,f(x_0+\varepsilon)) always contain some [x_0,l) for some l.

    As for the other direction, suppose for contradiction f is not continuous from ({\bf R}, \mathcal{F}_r) to ({\bf R}, \mathcal{F}). Then one can find an intervel (a,b) whose inverse cannot contain any intervel (also nonempty).Then we can find a x_0 with its image in (a,b) which is not right-continuous, thus a contradiction.

  10. PDEbeginner Says:

    Dear Xiaochuan,

    Thanks a lot for your reply!

    To be honest, I don’t quite understand your argument. But finally I figured out the proof.

    I think the key point for this exercise is to understand the open sets in ({\bf R}, \mathcal{F}_r): given any open set A in ({\bf R}, \mathcal{F}_r), if x \in A, then there exists some \delta>0 such that [x,x+\delta ) \subset A. With this observation, I think the proof is quite easy.

  11. PDEbeginner Says:

    There seems some formula not being shown in the above post, which is ‘[x, x+\delta) is the subset of A‘.

  12. waterloo2005 Says:

    请问这本书有电子版的吗?

  13. liuxiaochuan Says:

    书还没有出版,可以链接到陶的博客上,直接打印。

  14. waterloo2005 Says:

    就是博文吧,不是pdf之类的吧

  15. waterloo2005 Says:

    听说Terry年仅7岁的时候就养成了读数学书,题目全做的习惯。想想自己曾经有过的错误想法,汗颜。从现在开始倒还不晚。

    现在有些数学书,课后的习题都是东拼西凑来的,和内容的呼应太差,用那章的知识就是高斯也作不出来。现在国内这样的书太多了。

  16. abc Says:

    在verycd上可以搜到的

  17. Qiang Says:

    Hi, Liu

    I’m a graduate in ECE and recently become quite interested in Real Analysis which I think is useful to my own research. However, when I tried to approach those exercises in Tao’s lecture notes, it’s so hard for me (apparently :)). Do you have solutions to those exercises ?

    Thank you, I quite enjoy your blog !

    • liuxiaochuan Says:

      Dear Qiang:

      Thank you for your comment. Sorry for the delay. I indeed did almost all the exercises. But I didn’t write them down. If you are interested in any specific problems, I would be more than happy to discuss them with you.

      • Qiang Says:

        Hi, Xiaochuang

        So glad to hear from you ! I have several questions concerning about Exercise 17 in lecture note 1 (Caratheodory). For i->ii, since for element set, m(A)=m^{*}(A), so m(A)=m^*(A)=m^*((A\bigcap E)\bigcup(A\bigcap E^c)), because apparently elementary set A is Lebesgue measurable, so A\bigcap E and A\bigcap E^c are measurable, therefore by finite additivity of Lebesgue outer measure, we get m^*((A\bigcap E)\bigcup(A\bigcap E^c))=m^*(A\bigcap E)+m^*(A\setminus E) as desired. For ii->iii, any box $B$ is also an elementary set with $m(B)=|B|$, so we get |B|=m^*(B\bigcap E)+m^*(B\setminus E). Can you give me hints on iii->i ?

        And also I don’t know how to prove part 2 and 3 of Exercise 13 :(

        Thanks


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