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Sunday, August 14, 2011

Workshop 結束

其實原名為 workshop in real analysis,最後 workshop 講唔到 integration,時間不足。我地有 7 次 meeting,每次用一至一個半鐘黎做 present,10 至 15 分鐘休息及將淨低嘅時間用黎講 notes。每次 workshop 為時三個鐘。已講 topic:



notes 喺每次 workshop 前準備,打下打下都 40 頁 notes。都整得幾辛苦,d theorem 用就用得多,從新 develop 返出黎都幾麻煩。any way 對我黎講得著唔係太多 :),因為 d 野都比較 basic。希望下個 summer 會有人幫我完成埋 integration 果 part 同埋 expand 返第一課嘅野 (compactness, total boundedness, completeness, connectedness, etc)。

Thursday, August 11, 2011

無聊之下嘅產物

喺 Royden (4th edition) p.452 有一 Theorem 將 locally compact Hausdorff space 有嘅 property 集埋一齊:



其中證明 (iii) 嘅主要工具係 Urysohn's lemma,但由 Urysohn's lemma 又可推出 Tietze Extension Theorem, 從 (iii) 我地應該可以做得更多。
Modification of (iii). Let  X be locally compact Hausdorff. If  \mathcal O is a neighborhood of a compact subset  K of  X, then the continuous function  f:K\to \mathbb R may be extended to a function  F\in C_c(X) for which  F vanishes outside  \mathcal O.
If  f is bounded, say  |f|\leq M for some  M>0, then the extension above can be chosen so that  |F|\leq M on  X.
Proof. As  K is compact and  \mathcal O\supseteq K, by (ii) of theorem 7 above there is an open  V such that  K\subseteq V\subseteq \overline{V}\subseteq \mathcal O with  \overline{V} compact. Once again by (ii) of  theorem 7 above there is an open  U such that
 K\subseteq U\subseteq \overline{U}\subseteq V\subseteq \overline{V}\subseteq \mathcal O.

Now we do our extension, as  \overline{V} is a compact Hausdorff space,  f is continuous (w.r.t. subspace topology) on  K, by the Tietze extension theorem there is a  \mathcal F\in C (\overline{V}) such that  \mathcal F|_{K}= f|_K, we extend  \mathcal{F} on  X\setminus \overline{V} by defining  \mathcal F|_{X\setminus \overline{V}}\equiv 0. The change of  \mathcal F between  \overline{V} and  X\setminus \overline{V} may not be continuous, we will try to ``smooth" this transition. As  K is compact and  U\supseteq K, by (iii) of theorem 7 there is a  \psi \in C_c(X) such that  0\leq \psi \leq 1,  \psi=1 on  K and  \psi=0 on  X\setminus U. Now we claim that the product of continuous functions  F:= \mathcal F \cdot \psi will do.

Clearly  \mathop{\mathrm{supp}} F \subseteq \mathop{\mathrm{supp}} \psi, hence  F has compact support. To show  F is continuous on  X we use the following fact:
Fact. Let  X=\cup_\alpha X_\alpha be a union of open subsets. Then  f:X\to Y is continuous if and only if the restrictions  f|_{X_\alpha}:X_\alpha \to Y are continuous, where  X_i has the subspace topology.
Proof. It follows from the observation that: For any subset  A of  Y,  f^{-1}(A)= \cup_{\alpha} (f|_{X_\alpha})^{-1}(A).\qed
Observe that both  X_1:=V and  X_2:=X\setminus \overline{U} are open,  X=X_1\cup X_2. It is enough to argue  F|_{X_i}'s are continuous. On  X_1, since  \mathcal F is a continuous function on  \overline{V},  \mathcal F|_V is therefore a continuous function on  V. And as  \psi is continuous on  X, so  F|_{X_1} is continuous. On  X_2, since  \psi|_{X\setminus U}\equiv 0  \implies  \psi|_{X\setminus \overline{U}} \equiv 0, and thus  F|_{X\setminus \overline{U}} \equiv 0, hence  F|_{X_2} is continuous on  X_2. We also note that  F|_{X\setminus \mathcal O}\equiv 0.

When  |f|\leq M on  K, we repeat the proof above but that time  \mathcal F can be chosen such that  |\mathcal F|\leq M by the following version of Tietze extension theorem.\qed

建立呢種 extension 嘅原因係為左證明 Lusin's theorem on  (X,\mathcal B(X),\mu),其中  X 為 locally compact Hausdorff,  B(X) 為 Borel  \sigma-algebra on  X  \mu 為 Radon measure (Royden's definition: A Borel measure such that Borel set is outer regular and open set is inner regular),我嘅 approach (某習題) 係先證明 Lusin's theorem 對 simple function 成立,從而利用 simple functions  \{\phi_n\},  \phi_n\to f pointwise 及 Egoroff's theorem 及再利用上述 extension 完成證明 (已證明若  E\in \mathcal B(X),\mu(E)<\infty,那麼  E 是 inner regular)。

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Some problem for entertainment:
Problem. Let  f:\mathbb{R}\to\mathbb{R} be a differentiable function so that  \displaystyle\left|f(x)-\sin(x^2)\right|\le\frac{1}{4} for any  x\in\mathbb{R}. Prove that there exists a sequence of real numbers  \{x_n\}_{n=1}^\infty for which  \lim_{n\to\infty} f'(x_n)=+\infty .

Monday, August 1, 2011

見工失敗

前排去左樂善堂余近卿中學面試去做份做八日就賺到五千嘅暑期班,貌似係為考試唔合格嘅學生而設嘅補底班黎 (亦即係我冇乜機會發揮嘅班)。咁好喇我自己又冇乜點見過工,見步行步。去到自我介紹,我講唔夠一分鐘就收口,深知不妙,最終不獲聘收埸。

我估最大原因係我冇乜教學經驗。原因係,首先我係該校校友推薦;其次係請人果位老師係 UST 師姐;再者,我個樣都算平易近人丫...。但間間學校都係以經驗為優先嘅話咁我邊鬼有經驗喎...。同埋我都唔算冇教學經驗,不過對像係班 UG year 1 ...。