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math104-s21:s:martinzhai [2021/09/01 12:20]
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     * __Pointwise Convergence of Sequence of Functions__: Given a sequence of functions fn∈f_n \isin Map$(\Reals, \Reals),wesay, we say f_nconvergeto converge to fpointwiseif pointwise if \forall x\isin\Reals$, lim⁡n→∞fn(x)=f(x)  ⟺  lim⁡n→∞∣fn(x)−f(x)∣=0\lim_{n\to\infty} f_n(x) = f(x) \iff \lim_{n\to\infty} \lvert f_n(x) - f(x) \rvert = 0.     * __Pointwise Convergence of Sequence of Functions__: Given a sequence of functions fn∈f_n \isin Map$(\Reals, \Reals),wesay, we say f_nconvergeto converge to fpointwiseif pointwise if \forall x\isin\Reals$, lim⁡n→∞fn(x)=f(x)  ⟺  lim⁡n→∞∣fn(x)−f(x)∣=0\lim_{n\to\infty} f_n(x) = f(x) \iff \lim_{n\to\infty} \lvert f_n(x) - f(x) \rvert = 0.
       * Examples：Running and Shrinking Bumps.       * Examples：Running and Shrinking Bumps.
-{{ math104-2021sp:s:img_d0796d4b49e0-1.jpeg?400 |}}+{{ math104-s21:s:img_d0796d4b49e0-1.jpeg?400 |}}
  
 === Lecture 18 (Mar 18) - Covered Rudin Chapter 7 === === Lecture 18 (Mar 18) - Covered Rudin Chapter 7 ===
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       * Example: [10,20]⊂R[10, 20]\subset \Reals, then a partition would be P={10,15,18,19,20}P = \{10, 15, 18, 19, 20\}.       * Example: [10,20]⊂R[10, 20]\subset \Reals, then a partition would be P={10,15,18,19,20}P = \{10, 15, 18, 19, 20\}.
     * __U(P,f) and L(P,f)__: Given f:[a,b]→Rf:[a,b]\to \Reals bounded, and partion p={x0≤x1≤...≤xn}p = \{x_0 \leq x_1 \leq ... \leq x_n\}, we define U(P,f)=∑i=1nΔxiMiU(P,f) = \sum_{i=1}^{n} \Delta x_i M_i where $M_i= \sup \{f(x), x\isin [x_{i-1},x_i]\}$; L(P,f)=∑i=1nΔximiL(P,f) = \sum_{i=1}^{n} \Delta x_i m_i where $m_i= \inf \{f(x), x\isin [x_{i-1},x_i]\}$.     * __U(P,f) and L(P,f)__: Given f:[a,b]→Rf:[a,b]\to \Reals bounded, and partion p={x0≤x1≤...≤xn}p = \{x_0 \leq x_1 \leq ... \leq x_n\}, we define U(P,f)=∑i=1nΔxiMiU(P,f) = \sum_{i=1}^{n} \Delta x_i M_i where $M_i= \sup \{f(x), x\isin [x_{i-1},x_i]\}$; L(P,f)=∑i=1nΔximiL(P,f) = \sum_{i=1}^{n} \Delta x_i m_i where $m_i= \inf \{f(x), x\isin [x_{i-1},x_i]\}$.
-{{ math104-2021sp:s:img_0a0c56a64ed8-1.jpeg?400 |}}+{{ math104-s21:s:img_0a0c56a64ed8-1.jpeg?400 |}}
     * __U(f) and L(f)__: Define U(f)=inf⁡PU(P,f)U(f) = \inf_{P} U(P,f) and L(f)=sup⁡PL(P,f)L(f)= \sup_{P} L(P,f).     * __U(f) and L(f)__: Define U(f)=inf⁡PU(P,f)U(f) = \inf_{P} U(P,f) and L(f)=sup⁡PL(P,f)L(f)= \sup_{P} L(P,f).
       * Since ff is bounded, hence ∃m,M∈R\exists m,M\isin \Reals such that m≤f(x)≤Mm\leq f(x)\leq M for all x∈[a,b]x\isin [a,b], then ∀P\forall P partition of [a,b][a,b], U(P,f)≤∑i=1nΔxiM=M(b−a)U(P,f) \leq \sum_{i=1}^{n} \Delta x_i M = M(b-a), and L(P,f)≥m(b−a)L(P,f) \geq m(b-a), and m(b−a)≤L(P,f)leqU(P,f)≤M(b−a)m(b-a)\leq L(P,f) leq U(P,f) \leq M(b-a).       * Since ff is bounded, hence ∃m,M∈R\exists m,M\isin \Reals such that m≤f(x)≤Mm\leq f(x)\leq M for all x∈[a,b]x\isin [a,b], then ∀P\forall P partition of [a,b][a,b], U(P,f)≤∑i=1nΔxiM=M(b−a)U(P,f) \leq \sum_{i=1}^{n} \Delta x_i M = M(b-a), and L(P,f)≥m(b−a)L(P,f) \geq m(b-a), and m(b−a)≤L(P,f)leqU(P,f)≤M(b−a)m(b-a)\leq L(P,f) leq U(P,f) \leq M(b-a).
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 ==== Questions ==== ==== Questions ====
-  - {{math104-2021sp:s:img_769dec51bb88-1.jpeg?400|}} I understand the visualization of this recursive sequence, but  to 5\sqrt{5}?+  - {{math104-s21:s:img_769dec51bb88-1.jpeg?400|}} I understand the visualization of this recursive sequence, but  to 5\sqrt{5}?
   - In general, how to prove a set is infinite(in order to use theorem 11.2 in Ross)?   - In general, how to prove a set is infinite(in order to use theorem 11.2 in Ross)?
   - Is there a way/analogy to understand/visualize the closure of a set?   - Is there a way/analogy to understand/visualize the closure of a set?
   - Is there a way to actually test if a set is compact or not instead of merely finding finite subcovers for all open covers?   - Is there a way to actually test if a set is compact or not instead of merely finding finite subcovers for all open covers?
   - Rudin 4.6 states that if p∈Ep\isin E, a limit point, and ff is continuous at pp if and only if lim⁡x→pf(x)=f(p)\lim_{x\to p} f(x) = f(p). Does this theorem hold for p∈Ep\isin E but not a limit point of EE?   - Rudin 4.6 states that if p∈Ep\isin E, a limit point, and ff is continuous at pp if and only if lim⁡x→pf(x)=f(p)\lim_{x\to p} f(x) = f(p). Does this theorem hold for p∈Ep\isin E but not a limit point of EE?
-  - {{math104-2021sp:s:img_0324.jpg?400|}} How is the claim at the bottom proved?+  - {{math104-s21:s:img_0324.jpg?400|}} How is the claim at the bottom proved?
   - Could we regard the global maximum as the maximum of all local minimums?   - Could we regard the global maximum as the maximum of all local minimums?
   - Under what circumstance would Taylor Theorem be essential to our proof, since for the question appeared in HW11 Q2, I really do not think using Taylor Theorem is a better way to prove the claim?   - Under what circumstance would Taylor Theorem be essential to our proof, since for the question appeared in HW11 Q2, I really do not think using Taylor Theorem is a better way to prove the claim?
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   - Is there any covering for a compact set that satisfies total length of the finite subcovers for the set is less than ∣b−a∣\lvert b-a \rvert? (Answer is no)   - Is there any covering for a compact set that satisfies total length of the finite subcovers for the set is less than ∣b−a∣\lvert b-a \rvert? (Answer is no)
   - Question 16 on Prof Fan's practice exam.   - Question 16 on Prof Fan's practice exam.
-  - This is my solutions towards the practice exam: {{ math104-2021sp:s:practice_solutions.pdf |}}+  - This is my solutions towards the practice exam: {{ math104-s21:s:practice_solutions.pdf |}}
math104-s21/s/martinzhai.1630524022.txt.gz · Last modified: 2021/09/01 12:20 by 110.249.201.35

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