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)。
Sunday, August 14, 2011
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。
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:
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 及 \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)。
**********
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 .
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.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
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.
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).\qedObserve 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 及 \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)。
**********
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 ...。
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 ...。
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