Workshop Notes

最後一次整理 workshop notes 是從 UG year 3 到 PG year 1 的暑假。最近懂得用 mtpro2 這字體，便從新再換一下格式：

http://ihome.ust.hk/~cclee/document/WorkshopinAnalysis.pdf

若我能如期在暑假前畢業，可能會再多開一次 workshop 去說說 complex analysis --- 集合 real, functional 及 harmonic analysis 的一門優美學科。

weak $C^1$ norm dominates $C^0$ norm on $C^1[0,1]$ (to be used in tutorial).

The weak $C^k$ norm for $f:[a,b] \to \R$ is defined by \[
\|f\|_{W^{k,1}} =\sum_{i=0}^k \int_a^b |f^{(i)}(t)|\,dt.
\] This is an exercise (a problem stated in some forum) to show that the weak $C^1$ norm (hence weak $C^k$ norm, this concept comes from Sobolev spaces) dominates the $C^0$ norm (i.e., sup-norm):

Problem. If $f\in C^1[0,1]$, show that \[
\sup_{x\ge\in [0,1]}|f(x)|\leq \int_0^1|f(t)|\,dt +\int_0^1 |f'(t)|\,dt.
\] 

Solution. Denote $I=[0,1]$. Let $x_0\in I$ be such that $|f(x_0)|=\|f\|_{[0,1]}$. Then for any $\alpha,\beta\in I$ we have \[
\int_{x_0}^\beta |f'(t)|\,dt \ge |f(x_0)-f(\beta)|
\] and \[
\int_{\alpha}^{x_0} |f'(t)|\,dt \ge |f(x_0)-f(\alpha)|.
\] On summing up, by triangle inequality we have \[
\int_0^1|f'(t)|\,dt \ge \int_\alpha^\beta |f'(t)|\,dt \ge |2f(x_0)-f(\alpha)-f(\beta)|.
\] On the other hand, by continuity of $f$ and the differentiability of $g(x):=\int_0^x|f(t)|\,dt$ on $(0,1)$ we have \[
\int_0^1|f(t)|\,dt =g(1)-g(0)=g'(c)=|f(c)|,
\] for some $c\in (0,1)$. Therefore by triangle inequality again we have \begin{align*}
\|f\|_{W^1[0,1]}&:= \int_0^1|f(t)|\,dt+\int_{0}^{1} |f'(t)|\,dt\\
&\ge  |2f(x_0)-f(\alpha)-f(\beta)| + |f(c)|\\
&\ge |2f(x_0)-f(\alpha)-f(\beta)+f(c)|.
\end{align*} Now we set $\alpha = x_0$ and $\beta=c$ to get $\|f\|_{W^1[0,1]}\ge |f(x_0)|=\|f\|_{[0,1]}$, as desired.$\qed$

教授安裝 mathtime pro 2 字型

tex stackexchange 那有教授超簡單的使用方法：
http://tex.stackexchange.com/questions/110894/how-do-i-install-mtpro2

\pdfmapfile{=mtpro2.map}

在 office 也能夠輕鬆使用，不用安裝！(以前為安裝這字體弄得死去活來，耗費不少青春！)

某文件

在接着的學期我將要教 Math2033 的 tutorial (即 201)，instructor 由教了好幾年的何漢明轉為 Dr kin li。Math 2031 (即 202) 暫定不會再出現。

kin li 的 tutorial 對非常懶的 TA 來說很好，kin li 不會要求你在 tutorial 教甚麼，只須坐在課室聽學生做每周安排的 presentation 便何。但在我以至我前兩屆師兄的年代 (kin li 的) Math202/301 tutorial 不會單純聽 presentation，會用盡剩餘的時期多講講要知的 concept 和例子。為了延續這份精神，我不曾浪費掉 Math301 tutorial 多出的時間，這次的 Math2033 當然也不例外。因此我想起這份 pdf：

http://ihome.ust.hk/~cclee/document/Questions.pdf

中學時期的我便會使用 tex。那時很迷 tex，typeset 的數式很美，所以非常勇躍去解有趣的數學題，把它的解打出來，然後收錄。這習慣只維持到大學二年級。百多頁的文件處理起來不容易。就算現在懂得把 tex file 局部處理再合併 (tex 裏面做) 也懶得再整理這文件了。因我解的題愈來愈多，把我大學三年級到 mphil year 2 裏我認為有趣的題都收錄的話這份文件恐怕頁數將超過 200 --- 不易整理。

但解過的題都總想記錄一下，現在的途徑多把題分散在自己的 tutorial notes 或在這 blog 裏。在以上連結裏 elementary analysis 的部分將會散落到 2033 的 tutorial notes 吧～ :)

呀，另外在某一年為 Math201 的學生炮製了某 list 問題：

http://ihome.ust.hk/~cclee/document/math201.pdf

Math3033 (2013-2014) Final Exam

This year the mean of the final exam is 35.51/100 and the SD is 12.71. To wit, answering the first two problems perfectly is already safe to get $\text{C+}/\text{B-}$. If one does perfectly well in everything before the final exam (which are all very standard), then one gets an $\text{A-}$. This again reflects that there are too many weak students in Math3033 and getting an $\text{A-}$ is very easy for hard working students.

The first two problems are routine calculation, with one problem on uniform convergence and one on Lebesgue Dominated Convergence Theorem. The next 3 problems are as follows:

Problem 3. Show that the function $f:[0,1]\setminus \Q\to \R$ given by \[
f(x)=\sum_{i=1}^\infty \frac{a_i}{16^i},\quad\text{where } x=0.a_1a_2a_3\dots
\] is measurable.

Summary to Problem 3. Very few people can get it correct, some partial marks (up to 2) are given for those who try to prove $f$ is continuous, increasing, etc. Indeed this function is not continuous, any attempt proving this to be continuous just deserve very little marks. Some of the students claim the function is increasing, this still requires some manipulations of inequality and cannot be seen as trivial.$\qed$

Problem 4. Let $f:[0,1]\to[0,\infty)$ be measurable such that \[
\int_{[0,1]} f^n\, dm =c\in \R,\quad \forall n\in \N.
\] (a) Prove that $f^{-1}[2013,\infty)$ has measure zero.
(b) Prove that $f=\chi_E$ a.e. for some measurable set $E$.

Summary to Problem 4. By observing the answer in part (b), some students can guess the answer in part (a) but did not attempt to prove this. Actually it is very easy to show $m(f^{-1}(\alpha,\infty))=0$ for every $\alpha > 1$, therefore $f\leq 1$ a.e. The part (b) is hard to many students, they need to explain why $m(f^{-1}[0,1))=0$, this should not be difficult if one tries to play around with the term $\int_{[0,1]}f^n\,dm = c$ and split the integral into two parts. Then \[
m(f^{-1}(\{1\}))=1,
\] and therefore on $f^{-1}(\{1\})$ we have $f = 1=\chi_{f^{-1}(\{1\})}$. It follows that $f=\chi_E$ a.e., where $E=f^{-1}(\{1\})$.$\qed$

Problem 5. Let $T\subseteq [0,1]$ be a set of $7$ elements and $S\subseteq [0,1]$ be measurable such that $m(S)\ge \frac{1}{3}$, prove that there are $a,b\in T$ and $c,d\in S$ such that \[
|a-c|=|b-d|.
\] 

Summary to Problem 5. Only 2 people can get a decent amount of marks from me. Both of them try to prove by contradiction by supposing that $|a-c|\neq |b-d|$ for every $a,b\in T$ and $c,d\in S$. One of them can proceed successfully while another one is on the half way. Students usually work on equivalent statement of $|a-c|=|b-d|$ but never think of the possibility of having the sufficient condition: \[
a-c=b-d
\] for some $a,b\in T$ and $c,d\in S$. In this way one can find that \[
a+d=b+c.
\] Namely, we may try to prove $(a+S)\cap( b+S)\neq \emptyset$ for some $a,b\in T$. Suppose not, the we have the contradiction that $2\ge 7/3$.$\qed$

Conclusion. It seems that Dr Li thinks the midterm is easy enough to ensure most people can pass the course, therefore it tries to set the exam in a moderate level. Unfortunately very few students can develop a good background (absorb the material) and technique (methodology) in the basic measure theory.

Fourier Series

Usual exercises in Fourier series ask students to prove identity of the form \[
f=\sum a_n\cos nx+\sum b_n\sin nx.
\] Most of them are extremely straightforward and what you need is sufficient manpower, patience and time. If I were the one who teach Fourier series, I would propose the following identities as examples:

Problem. By using the Fourier series expansion of $f(x)=\cos \alpha x$, $x\in (-\pi,\pi)$, prove that \[
\cot x=\frac{1}{x}+\sum_{n=1}^\infty \brac{\frac{1}{x-n\pi}+\frac{1}{x+n\pi}}=\frac{1}{x}+\sum_{n=1}^\infty \frac{2x}{x^2-n^2\pi^2}
\] and also \[
\csc x=\frac{1}{x}+\sum_{n=1}^\infty (-1)^n \brac{\frac{1}{x-n\pi} +\frac{1}{x+n\pi}}=\frac{1}{x}+\sum_{n=1}^\infty (-1)^n \frac{2x}{x^2-n^2\pi^2}.
\] Also evaluate $\dis \sum_{n=1}^\infty \frac{1}{n^2-\alpha^2}$.

The surprise should be: there is apparently no evidence that Fourier series is involved!

Integral Inequality

Problem. Let $k$ be a positive integer.  $f:[0,\infty)\to[0,\infty)$ is a continuous function such that $f(f(x))=x^k$, $\forall x\in[0,\infty)$. Show that \[\int_0^1f(x)^2\,dx\ge\frac{2k-1}{k^2+6k-3}.\]

This statement is provable by Young's inequality. By assuming $f$ be continuously differentiable (or any kind of continuous such that the integration by parts on $[0,1]$ work, say $f$ is absolutely continuous), then we have a stronger bound: \[
\int_0^1f(x)^2\,dx\ge\frac{2k-1}{k^2+4k-2}.
\] Which becomes the previous one by adding $2k-1$ to the denominator.

Solution. There is a few observations. First, $f$ is injective. Second, $f(f(x))=x^k$ implies $f(f(f(x)))=f(x)^k$, therefore  $f(x^k)=f(x)^k$. In particular, if we put $x=0$ and $x=1$, then we have $f(0)=f(0)^k$ and $f(1)=f(1)^k$. Therefore both $f(0),f(1)\in \{0,1\}$.  By injectivity and the fact $f\ge 0$, we have $f(0)=0$ and $f(1)=1$.

By Cauchy-Schwarz inequality we have $I:=\int_0^1f^2\,dx\ge \int_0^1 f\,dx$. We substitute $x=f(y)$ to get \[
I=\int_0^1 f(f(y))(f(y))'\,dy=\int_0^1 y^k(f(y))'\,dy=1-k\int_0^1f(y)y^{k-1}.
\] The last equality follows from integration by parts. Now we shrink RHS by Cauchy-Schwarz inequality to get $I\ge1- \frac{k}{\sqrt{2k-1}}\sqrt{I}$, which we rearrange to get \[
(\sqrt{I})^2 +\frac{k}{\sqrt{2k-1}} \sqrt{I}-1\ge 0.
\] By completing square we find that this implies \[
\sqrt{I}\ge \frac{-\frac{k}{\sqrt{2k-1}}+ \sqrt{\frac{k^2+8k-4}{2k-1}}}{2}=\frac{2\sqrt{2k-1}}{\sqrt{k^2+8k-4}+k}.
\] Now by Cauchy-Schwarz inequality  we also have \[
\sqrt{k^2+8k-4}+k\leq \sqrt{(k^2+8k-4+k^2)(1+1)}=2\sqrt{k^2+4k-2},
\] hence we have \[
I\ge \frac{2k-1}{k^2+4k-2}.\qed
\]

Code

在 inline mdoe 時 matrix 縮細，display mode 時 matrix 為原本大小的方法：

msc 好恐怖

近排食下午茶撞到 Dr. Li，咁就一齊傾下計。佢話我知原來 Jimmy fung 又話我俾人投訴喺 msc (math support center) 到爆左 d 單字出黎。唔知係咪 Dr. Fung 親耳聽到，定係有人投訴，我自己就已經愈黎愈小心，亦都甚少落 msc (落親去都係搵人食飯，唔會逗留多過十分鐘)，都仲要咁岩俾人捉到又咁岩俾人投訴。話冇人針對我，你信唔信？但我自己又心虛，難以否認。

Dr. Li 叫我盡量將所有精神放喺數學，唔好諗其他野，咁就冇咁易有野俾人捉住黎講，仲叫我盡量做到冇野可以俾人拎黎講。咁有 d 難度喎...。

一條毒 L 行淘大商埸

以前淘大商場 3 樓周圍都係賣電現模型漫畫多，而家有好多地方都變左做賣電子產品、飾品及補習，仲要有 759 零食部 ...。

以前呢到一上樓梯就係賣遊戲/遊戲機，玩具，遊戲卡等。果時好多細佬渣住部 gameboy 聚喺塊玻璃後面搵人玩 pokemon 對戰 ...。

同一層，上圖所示嘅鋪位對面就係下面呢個景像，以前呢到前、左、右都係賣遊戲/動畫相關嘅野，而家變哂飾品/家具...。

material 準備完畢

今個 sem 準備左兩堆 material，到今個星期終於完成哂～。可以話下面係第 2 代嘅 notes。

Record some Problems

May be used in class of analysis:

Problem. Suppose $f'(x)<0<f''(x)$ for $x<a$; and $f'(x)>0>f''(x)$ for $x>a$. Prove that $f$ is not differentiable at $a$.  

Problem. If $f:\R\to \C^\times$ is a group homomorphism and is continuous, prove that $f(x)=e^{cx}$ for some $c\in \C$.

Problem. Let $C[a,b]$ be the ring of continuous functions on the finite closed interval $[a,b]\subseteq \R$.
(a) For $c\in [a,b]$, prove that $I_c$ given by \[I_c=\{f\in C[a,b]:f(c)=0\}\] is a maximal ideal of $c[a,b]$.
(b) Prove that every maximal ideal of $C[a,b]$ is $I_c$ for a unique $c\in [a,b]$.

Problem. Define a sequence by $a_1 = 1$, $a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$, if it exists.

Problem. Denote $A = \{ x \in \mathbb{R} : \lim_{n\to\infty} \{ 2^n x\} = 0 \}$. Prove that $m(A) = 0$. Is $A$ countable or uncountable?

Notation: $\{x\}$ denotes the fractional part of $x$, given by $\{x\} = x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the unique integer for which $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$.

Problem. Find $\dis  \prod_{k=0}^\infty(e^{a\cdot 2^{-k}}+e^{-a \cdot 2^{-k}}-1)$.

Problem. Find $\dis \lim_{n\to\infty}\left(\prod_{k=1}^{n}\cos{\frac{\pi k}{2n}}\right)^{\frac{1}{n}}$.

Problem.  Let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable ($C^1$) function such that $\dis f(x) = f(\frac{x}{2})+ \frac{x}{2} f'(x)$.

Show that $f(x)$ is a linear function, i.e. $\exists (a,b) \in \mathbb{R}^2$, $f(x)= ax+b$.

Problem. Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that \[\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0.\] Then prove that \[\lim_{N\to\infty}\int_0^\pi\frac{\sin(N+\frac12)x}xdx=\pi/2.\] Finally, prove that \[\int_0^\infty\frac{\sin x}xdx=\pi/2.\]

Problem. Let $a>0$ and $\{x_n\}$ a bounded sequence of real numbers. If $\{x_n\}$ doesn't converge and $\lim_{n \to \infty} (x_{n+1}+ax_n)=0$, prove that $a=1$.
Remark. We can generalize the problem by merely assuming $\lim_{n \to \infty} (x_{n+1}+ax_n)$ exists.

Problem. Let $f: \mathbb{R} \to (0,\infty)$ be a continuous differentiable function such that $f'(x)=f(f(x))$ for all $x\in \mathbb{R}$. Show that there is no such function. 

Problem. Suppose $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that, $|f(x)-f(y)| \ge \frac{1}{2}|x-y|$ for all $x,y \in \mathbb{R}$. Is $f$ necessarily one to one and onto?

Problem. Let $f:[0,1]\to \R$ be continuous and differentiable on $(0,1)$ such that $f(0)>0,f(\frac{1}{2})<0$ and $f(1)>1$. Show that there is a $\xi\in (0,1)$ such that $f'(\xi)=f(\xi)$. 

一些歷史遺物

在 photobucket 那尋回九張作品，暫時年份不明 (可以肯定每一張都在某論壇發表過，但懶得找了 ...)。

The application of Lagrange multiplier's method with multiple constraints to linear algebra

In this post we record an application of a standard result in multivariable calculus (whose rigorous proof is still beyond the standard multivariable calculus course) to proving the following standard fact in linear algebra:

Theorem (Spectral). Let $A$ be an $n\times n$ real symmetric matrix, then $A$ is orthogonally diagonalizable. 

In other words, there is an orthonormal  basis $\{v_1,\dots,v_n\}$ such that each of $v_i$'s is an eigenvector. Througout the proof the following will be assumed:

Theorem (Lagrange's Multiplier). Let $n\ge m$, suppose that:

• $G=(g_1,\dots,g_m):\R^n\to \R^m$ is $C^1$ on $\R^n$ and $G'(x_0)$ has full rank.
•  $f$ is a real-valued function defined near $x_0$ and is differentiable at $x_0$.
• $f$ has a  local extreme at $x_0$ under the constraint $G(x_0)=0$.

 Then there are $\lambda_1,\lambda_2,\dots,\lambda_m\in \R$ such that \begin{equation}\label{equality of lagrange}\nabla f(x_0)=\lambda_1\nabla g_1(x_0)+\lambda_2\nabla g_2(x_0)+\cdots +\lambda_m \nabla g_m(x_0).\end{equation}

A detailed proof of this multiplier method has been recorded in my lengthened version of Math3033 tutorial note 3.

Proof of Spectral Theorem. The maximum of \[
\psi (x) = x^TAx
\] w.r.t. the constraint $x^Tx=1$ can be attained since $S^{n-1}$ is compact. Now it is easy to show that \[\psi'(x)=2x^TA\quad \text{and}\quad \nabla(x^Tx)(x)=2x^T,
\] therefore the existence of the constrained extreme implies that there is $\lambda\in \R$ and a vector $v\in \R^n$ such that

1. $v^Tv=1$.
2. $\nabla \psi (v)=\lambda\nabla (x^Tx-1)(v)\iff 2v^TA=2\lambda v^T\iff Av=\lambda v$.

Therefore we have obtained our first eigenvalue and eigenvector.

Now in view of the statement of Spectral Theorem let's suppose that there are already $\lambda_1,\dots,\lambda_k\in \R$ and $v_1,\dots,v_k$ orthonormal such that $Av_i=\lambda_iv_i$ for each $i=1,\dots,k$. We try to find $\mu\in \R$ and $u\in S^{n-1}$ such that $Au=\mu u$ and $v_i\cdot u=0$ for all $1\leq i\leq k$. After that our proof is completed by induction. 

Following the same idea in the first paragraph of the proof we know that the constrained extreme problem with the following data can be attained: \[ x^TAx = \text{max},\quad x^Tx=1,\quad v_i^Tx=0,\forall i=1,\dots,k.
\] Therefore there are $\mu_1,\dots,\mu_k,\mu\in \R$ and a $u\in\R^n$ satisfying the constraints such that \[
\nabla  (x^TAx)(u)=  \mu \nabla(x^Tx-1)(u)+\sum_{i=1}^k \mu_i\nabla(v_i^Tx)(u),
\] on simplification we get \[
2u^TA=2 \mu u^T+\sum_{i=1}^k \mu_i v_i^T.
\] From our hypothesis in the last paragraph, by applying $v_i$'s on both sides above we have for $i=1,2,\dots,k$, \[
0=0+\mu_i \|v_i\|^2,
\] therefore $\mu_1=\mu_2=\cdots=\mu_k=0$, and we have \[
2u^TA=2\mu u^T\iff Au=\mu u.
\] Recall that $u$ satisfies the constraints $v_i\cdot u=0$ for $i=1,\dots,k$ and $u\cdot u=1$.$\qed$

重新開張

Wordpress 說起

大概從中五開始用 blog (xanga 時期)，那時亦開始間中打一些數學 post，到中七時為方便輸入數式而轉往 wordpress。在我認知裏，方便打 math 的 blog 只有 blogger 及 wordpress。而我一開始是試用 wordpress 的，因而沒有嘗試 blogger。

大約二年前我發覺在 wordpress.com 打數學文章異常辛苦，原因是每次打數式，如 $1+1=2$，都必須輸入

$\$\text{latex }1+1=2 \$$。

那種繁複程度磨滅了我打數學文章的意欲。直至上年見識到孟國武教授用 blogger 上 lecture，我才知道原來 blogger 可以用一套名為 mathjax 的 javascript typeset 漂亮的數式。這在 wordpress.com (直至現在) 是辦不到的，因它不讓你在 header 內放入任何 javascript。現在 blogger 打數學不用像從前 wordpress 時在每句數式上加上 ``latex'' 。

有了一年以 blogger 為 tutorial 的 blog 的經驗，我開始熟習在 blogger 輸入數式的「方法」。「方法」所指的是 spacing 的控制，因在 blogger 以平常用 latex 的方式輸入數式會容易造成過多的換行。

幸運地在網上很容易找到由 wordpress $\to$ exported xml $\to$ blogger xml $\to$ import to blogger 的方法，所以最近耗了半天將 wordpress 移植到 blogger，當中出現了很多錯誤 =.=。

現在在 UST 教 tutorial，某程度上正在 represent 數學系，所以不宜讓學生知道太多我過去的「祕密」，尤其是我以前所畫的畫對我的形像 (？) 有一定的影響，所以決定 private 這 blog 及只讓熟人看。

BLOG 定向

本想在這 blog 上記錄一些發洩性言論，但這些易得罪他人的評論少放上網為妙了 (雖然這 blog 是 private 的)，因難以判斷哪些身邊的人是女王的線人。所以這 blog 的主旨傾向自己的無謂「日常」及討論數學問題。一如以往，我以"題"為本，不會無謂地將一套 theory 重新在 blog 上打一次。

Record some Problems

In this post I want to record 3 problems:

The following is a problem in GRE math subject test:

Problem 1. Let $\F_p$ be a finite field of order $p$, show that the number of noninvertible $2\times 2$ matrices over $\F_p$ is \[p^4-p^3-p^2+p.\]

Can we generalize to $n\times n$ matrices?

The following is a problem found somewhere by someone in Taiwan's forum, actually I also find this in mainland textbook on real functions:

Problem 2. Let $E,E_1,E_2\subseteq \R$ be such that $E=E_1\cup E_2$. If $E$ is measurable such that \[
m(E)=m^*(E_1)+m^*(E_2)<\infty,
\] show that $E_1$ and $E_2$ are measurable.
HINT: Approximate $E_1,E_2$ from outside by measurable set.

The following is a problem in the midterm exam of Math4061 (fall 2013), which I want to record a simple proof here.

Problem 3. Let $X,Y$ be metric spaces. Suppose that $f:X\to Y$ is a function such that

• $f$ is 1-1.
• $K$ compact in $X$ $\implies$ $f(k)$ compact in $Y$.

Prove that $f$ is continuous.

Dr Li's original solution proves by contradiction, I want to give a direct proof using the sequential criterion of continuity for metric spaces.

Solution to Problem 3. Suppose that $x_n\to x$ in $X$, let's prove that $f(x_n)\to f(x)$ in $Y$. First we observe that \[
L:=\{x_n:n\ge 1\}\cup \{x\}
\] is compact. Second we define $g:=f|_{L}:L\to f(L)$ and show that $g^{-1}:f(L)\to L$ is continuous. Indeed, let $K\subseteq L$ be any closed set, since $L$ is compact, so is $K$, and thus \[
f(K)=g(K)=(g^{-1})^{-1}(K)
\] is a compact set, hence closed set, in $Y$ by hypothesis. Therefore any closed set in $L$ is pulled back to a closed set in $f(L)$, $g^{-1}$ is continuous. Since $f(L)$ is compact and $L$ is Hausdorff (as it is a metric space), a standard training in topology enables us to show $g^{-1}$ is a homeomorphism, so $g$ is continuous.

Finally we check that $f(x_n)\to f(x)$. Since $x_n\to x$ in $X$, we have $x_n\to x$ in $L$, so $g(x_n)\to g(x)$ in $f(L)$, but then $f(x_n)=g(x_n)\to g(x)=f(x)$ in $Y$, so we are done.$\qed$

Linear Algebra Notes 整合

最近將 winter 自製嘅 6 份 Linear Algebra Notes 合拼，並將 d proof 填上，用左兩日 (屋企``休息" 時間) 將 chapter 1 -- chapter 4 搞掂好哂。遲 d 有時間再搞埋 chapter 5，咁就會完成一份冇人會睇嘅書  == 。其實只係自我娛樂下同當係筆記咁將自己學過嘅野 summarize 返 (以前亦真係有人會印我以書形式製成嘅 notes 做參考架 ^^)。

整好：http://ihome.ust.hk/~cclee/document/MATH2121/Combine_Notes.pdf

完 sem！

又隔左幾個月冇更新呢個 blog。

今個學期真係好辛苦，自己又要 tu 五個 tutorial section，又要搞掂 d background reading (為左做 research)，同時又要做跨境學童過中大讀 real analysis II 同 pde-I (聽聞冇 II? ==)。基本上呢兩科都走左好多堂 (縱然 real analysis II 如 I  一樣教得非常好)。而教 PDE-I 嘅 instructor 完全唔俾 material，唔講進度。如果唔係有 fd 幫我做針我真係完全唔知邊 d 有講邊 d 冇講。PDE-I 辛苦嘅地方係冇人可以同我討論，自己一個磨。turns out 同上個 sem 一樣，theoretical PDE 讀多次都仲係唔太識 PDE。

有鑑於上個 sem 準備左太多野俾學生，以至上唔上 tutorial 都冇所謂，今個 sem 上堂用嘅 notes 都係最多 4 版紙 (i.e., 最多兩張紙)，其中有好多空間俾人寫野 (咁係因為俾太多野學生，根本唔知我想講嘅重點喺邊！)。基本上都係 problems, problems and problems，好少會打 basic theorems 出黎，因為解題時會講返。我另外建立左個 blog (例1, 例2) 去講返所有 solution，但可能冇咁詳細。我估因為呢種 approach 唔錯，85 學生嘅 course (math1014) keep 住有 50 -- 55 學生肯每個星期都露一露面，對我黎講已經好滿足 (我估佢地唔會因為間唔中出左 d 難題目而 e 死我嘅 ...)。

至於呢個 sem 嘅 math2121 (math111) 係正常 linear algebra course 嘅閹割版。唔教 sums，唔教 matrix representation，唯一嘅 inner product 只有 dot product (莫講話何謂 positive/negative definite ...) 。唔講得 set，冇 method of proof，全 computation。我估 prof. 孟喺 UST 教得太耐，深知 spring sem 通常有好多 nonmath major 同數底比較差嘅學生讀，所以故意講得比較慢同講少 d 內容。其實我都教得好無癮，因為我係照返正常 linear algebra 要教嘅野黎講 (我有向 prof 滙報架！)，不過只係得返 d 有能力同有心讀嘅學繼續堅持黎上堂。只可以慨嘆呢個 sem 好多學生可能只係想 pass 左呢個 course 而唔係想學好 linear algebra 呢門最基礎同最簡單嘅學科。

講到教書，有 physics TA 向我反映佢地有我嘅 linear algebra 學生連 matrix 乘落一支 column vector 都唔識乘。我心諗我冇義務喺 tutorial  重新再教一 d 學生本來就要自己睇同識嘅野。我嘅出發點係，只有做數學先可以學好數學，好嘅題目同埋好嘅 idea 係最最最重要，計算從來係只要想做就可以做得好嘅野。我所以通常除左最 basic 要知嘅 computation 以外，都會多講返多少少 linear algebra 裏面最重要嘅概念同數學上嘅論證手法。大學唔係中學，我覺得要降低教學內容到平庸水準去迎合學生係對好嘅學生非常之不合理。況且最 basic 要知嘅野我全部都有詳細咁講，學生吸唔吸收同肯唔肯去學完全唔係我嘅責任。

不過通常我呢 d 觀點都會俾同我對數學抱不同態度嘅人所批評。我識嘅某 d 應數學生同 physics 學生都覺得學 proof 好無謂，識 compute 就得。嗯 ...

下年開始終於轉返做 full-time MPhil，唔駛再做咁多教學同可以專心做我嘅 research。希望 math department 會體恤我呢個學期教左咁多，可以減少我下年嘅職務喇... T"T。

Code

The complete Times-change using mathptmx

完 sem 了

這個 sem 我負責 Math3033 (301) 及 math1018 的導修課。經歷了這個學期，我開 sem 前的那鼓教學熱誠已被徹底磨滅，其中主要的原因是 math3033。

對於 Math3033，我自身從沒有上過這一門課 (我上的是 3043 (204))，所以也沒弄清楚學生 linear algebra 要透澈到甚麼程度。第一堂根據過往的 TA 做法，應當是 review linear algebra。上堂前我也問清楚 kin li (這門課的 instructor) linear algebra 要 review 到甚麼程度，他說：「講返咩係 linear transformation，點樣 choose basis 同 form matrix 就得架喇。」嗯，這沒啥，很基礎的 linear algebra 內容，所以我把 basis，linear transformation 及它的 matrix w.r.t. 某些 basis 講一篇。學生開始亂了 (我第一節課也順帶提一提在 $  \mathbb R^n$ norm 有很多種，但他們是``等價的")，而一些認真的內地的同學問 linear algebra 對這門課是不是非常重要。

有見及此，我接着在 tutorial 盡量把 linear algebra 的內容有多──少說多──少。第一堂 tutorial 已經把不少同學嚇走了，接着的第二第三第 $  \cdots$ 節 tutorial 人數驟降及收斂得很快 (幸而減去 presentation 後立刻離開的一群，這個數字是非 0 的)。到最後一次 tutorial 有 7 人 (除去 present 後即走的人)。

不曉得這個 sem 我出了甚麼問題，是我教的東西太沒意思，太簡單嗎？

對於有意認真學習實分析的同學：Math3033 這門課只有後半學期的內容才是外面的學校 ``實分析" 所涵蓋的用容，所以這門課可以學到的測度論知識實在太少 (老實說，在 UST 讀 3033 這門分析課基本上都學不到實分析裏重要的思想)。建議真的有興趣的學生把 HL Royden, Real Analysis 這本書裏的 part I 及 part III 學好。然後可以找一本書把 $  \mathbb R^n$ 的 measure theory 學好。最後就是多做問題，多看看那些定理的應用 (這也是我 tutorial note 的宗旨，給出了所有 exercise 的解答)。

我的 tutorial notes：
https://sites.google.com/site/mathcclee/past-courses/math3033
我會經常微調這些 notes (因為我太閒，有時看到值得修改的地方便會修改)。當我看到有趣及合理的問題也會放進 exercise 裏。你也可以嘗試裏面的問題然後和我討論，也可以作為挑戰給我一些有趣的題。