Merge branch 'main' of https://github.com/Trance-0/NoteNextra

pages/Math4111/Math4111_L2.md

@@ -61,31 +61,31 @@ WLOG $\alpha>\beta$ and $\beta>\alpha$.
 
 EOP
 
-We write $SupE$ to denote the LUB of $E$.
+We write $\sup E$ to denote the LUB of $E$.
 
 This also applies to $GLB$ (greatest lower bound) and infinum of $E$
 
-#### Example 
+Example:
 
 1. $S=\mathbb{Q}, E=\{1,2,3\}$ ($E$ is bounded above)
-    * $SupE=3$, $Inf E=1$
+    * $\sup E=3$, $\inf E=1$
 2. $S=\mathbb{Q}, E=\{x\in \mathbb{Q}:0<x<1\}$  ($E$ is bounded above)
-    * $SupE=3$, $Inf E=1$
+    * $\sup E=3$, $\inf E=1$
 
-$SupE$ and $Inf E=1$ don't have to $\in E$
+$\sup E$ and $\inf E=1$ don't have to $\in E$
 
 3. $S=\mathbb{Q}, E=\{x\in \mathbb{Q}:0<x\}$ ($E$ is not bounded above)
-   * $SupE=\infty$ or not defined, $Inf E=0$
+   * $\sup E=\infty$ or not defined, $\inf E=0$
 4. $S=\mathbb{Q}, E=\phi$.
-   * $SupE=-\infty$ or not defined, $Inf E=\infty$ or not defined, we don't put $\infty$ in $\mathbb{Q}$
+   * $\sup E=-\infty$ or not defined, $\inf E=\infty$ or not defined, we don't put $\infty$ in $\mathbb{Q}$
 
 Important example
 
 5. $S=\mathbb{Q}, A=\{p\leq \mathbb{Q}:p>0, p^2<2\}$.
-    * $A$ is not empty and bounded above. However, $Sup A$ des not exists.
+    * $A$ is not empty and bounded above. However, $\sup A$ des not exists.
 
 If $S=\mathbb{R}, A=\{p\leq \mathbb{Q}:p>0, p^2<2\}$.
-    * $A$ is not empty and bounded above. However, $Sup A=\sqrt{2}$.
+    * $A$ is not empty and bounded above. However, $\sup A=\sqrt{2}$.
 
 #### Least upper bound property (LUBP)
 
@@ -93,7 +93,7 @@ if $\forall E\subset S$ that tis non-empty and bounded above, $\exist Sup E\in S
 
 #### Greatest upper bound property (GLBP)
 
-S has greatest lower bound property (GLBP) if $\exist E\subset S$ that is non-empty and bounded below, $\exists Inf E\in S$
+S has greatest lower bound property (GLBP) if $\exist E\subset S$ that is non-empty and bounded below, $\exists \inf E\in S$
 
 $\mathbb{Q}$ does not have LUBP and GLBP.
 

pages/Math4111/Math4111_L3.md

@@ -4,7 +4,7 @@
 
 Let $S=\mathbb{Z}$.
 
-1. Let $E=\{x\in S:x>0,x^2<5\}$. What are $sup\ E$ and $inf\ E$?
+1. Let $E=\{x\in S:x>0,x^2<5\}$. What are $sup\ E$ and $\inf\ E$?
   
     $sup\ E=2,inf\ E=1$
 
@@ -14,11 +14,11 @@ Let $S=\mathbb{Z}$.
 
 3. Does $S$ have the least upper bound property?
 
-    Yes, $\forall E\subset S$ that tis non-empty and bounded above, $\exist Sup E\in S$.
+    Yes, $\forall E\subset S$ that tis non-empty and bounded above, $\exist \sup E\in S$.
 
 4. Does $S$ have the greatest lower bound property?
 
-    Yes, $\forall E\subset S$ that tis non-empty and bounded below, $\exist Inf E\in S$.
+    Yes, $\forall E\subset S$ that tis non-empty and bounded below, $\exist \inf E\in S$.
 
 ## Continue
 
@@ -26,9 +26,9 @@ Let $S=\mathbb{Z}$.
 
 Proof that $LUBP\implies GLBP$.
 
-Let $S$ be an ordered set with LUBP. Let B<S be non-empty and bounded below.
+Let $S$ be an ordered set with LUBP. Let $B<S$ be non-empty and bounded below.
 
-Let $L=y\in S:y$ is a lower bound of B$\}$. From the picture, we expect $sup\ L=inf\ B$ First we'll show $sup\ L$ exists.
+Let $L=y\in S:y$ is a lower bound of $B$. From the picture, we expect $\sup L=\inf B$ First we'll show $\sup L$ exists.
 
 1. To show $L\neq \phi$.
 
@@ -37,7 +37,7 @@ Let $L=y\in S:y$ is a lower bound of B$\}$. From the picture, we expect $sup\ L=
 
     $B$ is not empty $\implies \exists x\in B\implies x$ is a upper bound of $L$.
 
-3. Since $S$ has the least upper bound property, $sup L$ exists (in $S$).
+3. Since $S$ has the least upper bound property, $\sup L$ exists (in $S$).
 
 Let's say $\alpha=sup\ L$. We claim that $\alpha=inf\ B$. We need to show $2$ things.
 
@@ -79,11 +79,11 @@ Remark:
 1. It's more helpful if you try to prove these yourselves. The proofs are "straightforward".
 2. For this course, it's not important to remember which properties are axioms, etc.
 
-Example of proof: 
+Example of proof:
 
 #### 1.14(a) $x+y=x+z\implies y=z$
 
-Proof: 
+Proof:
 
 $x+y=x+z$,
 

pages/Math4111/Math4111_L4.md

@@ -2,7 +2,7 @@
 
 ## Review
 
-1. Let $F$ be a field. Let $a,b,c,...,z\in F$ . Using he field axioms, simplify 
+1. Let $F$ be a field. Let $a,b,c,...,z\in F$ . Using he field axioms, simplify
 
     $$
     (x-a)(x-b)(x-c)...(x-z)
@@ -10,7 +10,7 @@
 
     $x\in F$, it must be at least one $0$ in the product...
 
-2. Suppose $A,B\subset\mathbb{R}$. Suppose $A$ and $B$ are nonempty and bounded above,$A\subset B$. WHat can you say about $sup\ A$ and $sup\ B$? Please justify.
+2. Suppose $A,B\subset\mathbb{R}$. Suppose $A$ and $B$ are nonempty and bounded above,$A\subset B$. WHat can you say about $\sup A$ and $\sup B$? Please justify.
 
     $$
     \forall x\in A, x\in B.  sup\ A\leq sup\ B
@@ -31,7 +31,7 @@ Proof
 
 Suppose the property is false, then $\exist x,y\in \mathbb{R}$ with $x>0$ such that $\forall v\in \mathbb{N}$, nx\leq y$
 
-Let $A=\{nx:n\in\mathbb{N}\}$. Then $A\neq\phi$ (Since $x\in A$) and $A$ is bounded above by $y$. Since $\mathbb{R}$ has LUBP, $sup\ A$ exists. Let $\alpha=sup\ A$.
+Let $A=\{nx:n\in\mathbb{N}\}$. Then $A\neq\phi$ (Since $x\in A$) and $A$ is bounded above by $y$. Since $\mathbb{R}$ has LUBP, $sup\ A$ exists. Let $\alpha=\sup A$.
 
 $x>0\implies \alpha-x<\alpha$, $\alpha-x$ is not an upper bound of $A$. (Since $\alpha$ is the LUB of $A$) $\implies \exist m\in \mathbb{N}$ such that $mx>\alpha-x$ by definition of $A$.
 

pages/Math4111/Math4111_L8.md

@@ -10,7 +10,7 @@ It should be empty. Proof any point cannot be in two balls at the same time. (By
 
 ### Metric space defs
 
-1. $p\in X,r>0$, $B_r(p)=\{q\in X:d(p,q)<0\}$, also called **neighborhood**.
+1. $p\in X,r>0$, $B_r(p)=\{q\in X:d(p,q)<r\}$, also called **neighborhood**.
 2. $p$ is a **limit point** of $E(p\in E')$ if $\forall r>0$, $(B_s(p)\cap E)\backslash \{p\}\neq \phi$
 3. If $p\in E$ and $p$ is not a limit point of $E$, then $p$ is called an **isolated point** of $E$.
 4. $E$ is **closed** if $E'\subset E$

pages/Math4121/Math4121_L13.md

@@ -1,5 +1,69 @@
 # Math4121 Lecture 13
 
+## Hidden Chapter 1
+
+This chapter is not covered in the lecture but I still want to mention it here.
+
+At first, when the integral was first invented, it was thought to be the area under the curve or above the curve, using intuitive geometric definition from the mysterious common sense of the homo-sapiens. There was not a rigorous definition of the integral from the eighteenth century, when it was first invented, to the nineteenth century, when Riemann, Lebesgue, and others rigorously defined the integral.
+
+The integral was thought to be the anti-derivative, for the general publics.
+
+However, we want to apply the integral to more general functions, rather than just the differentiable functions.
+
+So, we need a rigorous definition of the integral, one potential solution is the Cauchy-Riemann integral.
+
+### Riemann integral
+
+Recall from the previous lectures, we have the following definition of the Riemann integral:
+
+A function $f$ is Riemann integrable on $[a,b]$ if there exists a number $V$ such that for every $\epsilon>0$, there exists a $\delta>0$ such that for every partition $P=\{x_0=a,x_1,\cdots,x_n=b\}$ of $[a,b]$ with mesh less than $\delta$, we have
+
+$$
+\left|\sum_{i=1}^{n}f(x_i^*)(x_i-x_{i-1})-V\right|<\epsilon
+$$
+
+where $x_i^*$ is a point in the $i$-th subinterval $[x_{i-1},x_i]$.
+
+This sum only exists if the Darboux's sum defined by the following is small:
+
+### Darboux's sum
+
+Let $M_i=\sup_{x\in [x_{i-1},x_i]}f(x)$ and $m_i=\inf_{x\in [x_{i-1},x_i]}f(x)$.
+
+Then, the Darboux's sum is defined as
+
+$$
+\underline{S}(f,P)=\sum_{i=1}^{n}m_i(x_i-x_{i-1})
+$$
+
+and
+
+$$
+\overline{S}(f,P)=\sum_{i=1}^{n}M_i(x_i-x_{i-1})
+$$
+
+In this case, small means that $\forall \epsilon>0$, there exists a $\delta>0$ such that if $x_i-x_{i-1}<\delta$, then
+
+$$
+\sum_{i=1}^{n}(M_i-m_i)(x_i-x_{i-1})<\epsilon
+$$
+
+$(M_i-m_i)$ is the oscillation of $f$ on the $i$-th subinterval $[x_{i-1},x_i]$.
+
+#### Theorem 2.1: Riemann's Integrability Criterion (corollary version)
+
+A function $f$ is Riemann integrable on $[a, b]$ if and only if for every $\sigma>0$ be the bound for the oscillation of $f$, and for any $\epsilon>0$, we can find a subinterval length $\delta$, such that for any partition $P$ of $[a, b]$ with each subinterval has length less than $\delta$, the length of the sum of the lengths of the subintervals where the oscillation exceeds $\sigma$ is less than $\epsilon$.
+
+That is, mathematically, $\forall \sigma,\epsilon>0$, a function $f$ is Riemann integrable on $[a, b]$ if there exists $\delta>0$ such that $\forall P=\{x_0=a,x_1,\ldots,x_n=b\}$ where $x_i-x_{i-1}<\delta$.
+
+$$
+\sum_{x_i\in J}\Delta x_i<\epsilon,\quad \text{where} \quad J=\{x_i|(M_i-m_i)>\sigma\}
+$$
+
+#### Theorem 2.2: Darboux Integrability Condition
+
+Let $f$ be a bounded function on $[a,b]$. This function is Riemann integrable on $[a, b]$ if and only if for every $\epsilon > 0$, there exists a partition $P$ of $[a, b]$ such that the upper sum $\overline{S}(P;f)-\underline{S}(P;f)<\epsilon$
+
 ## New book Chapter 2
 
 Riemann's motivation: Fourier series

pages/Math4121/Math4121_L14.md

@@ -2,7 +2,7 @@
 
 ## Recap
 
-### Hankel developed Riemann's integrabilty criterion.
+### Hankel developed Riemann's integrability criterion
 
 #### Definition: Oscillation
 
@@ -22,9 +22,13 @@ $$
 
 where $\mathcal{P}=\{i:\omega(f,I_i)>\sigma\}$.
 
-Proof as homework questions.
+Proof:
 
-#### Corollary 2.4
+To prove Riemann's Integrability Criterion, we need to show that a bounded function $f$ is Riemann integrable if and only if for every $\sigma, \epsilon > 0$, there exists a partition $P$ of $[a, b]$ such that the sum of the lengths of the intervals where the oscillation exceeds $\sigma$ is less than $\epsilon$.
+
+EOP
+
+#### Proposition 2.4
 
 For point $c\in[a,b]$, define the oscillation at $c$ as
 
@@ -54,7 +58,7 @@ $$
 c_e(S) = \inf_{c\in C_s}\ell(C)
 $$
 
-where $\C_s$ is the set of all finite covers of $S$.
+where $C_s$ is the set of all finite covers of $S$.
 
 Example: 
 

pages/Math4121/Math4121_L16.md

@@ -73,7 +73,7 @@ A set is uncountable if it is not countable.
 
 #### Theorem: $\mathbb{R}$ is uncountable
 
-Easy proof using[Cantor's diagonal argument](https://notenextra.trance-0.com/Math4111/Math4111_L6#theorem-214).
+Easy proof using [Cantor's diagonal argument](https://notenextra.trance-0.com/Math4111/Math4111_L6#theorem-214).
 
 A new one
 

pages/Math4121/_meta.js

@@ -16,18 +16,10 @@ export default {
     Math4121_L11: "Introduction to Lebesgue Integration (Lecture 11)",
     Math4121_L12: "Introduction to Lebesgue Integration (Lecture 12)",
     Math4121_L13: "Introduction to Lebesgue Integration (Lecture 13)",
-    Math4121_L14: {
-        display: 'hidden'
-    },
-    Math4121_L15: {
-        display: 'hidden'
-    },
-    Math4121_L16: {
-        display: 'hidden'
-    },
-    Math4121_L17: {
-        display: 'hidden'
-    },
+    Math4121_L14: "Introduction to Lebesgue Integration (Lecture 14)",
+    Math4121_L15: "Introduction to Lebesgue Integration (Lecture 15)",
+    Math4121_L16: "Introduction to Lebesgue Integration (Lecture 16)",
+    Math4121_L17: "Introduction to Lebesgue Integration (Lecture 17)",
     Math4121_L18: {
         display: 'hidden'
     },