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math104-s21:s:ryotainagaki [2021/05/12 08:42]
73.15.53.135 		math104-s21:s:ryotainagaki [2026/02/21 14:41] (current)
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 1. Sets and sequences. (Ch 1- 10 in Ross) \\ 1. Sets and sequences. (Ch 1- 10 in Ross) \\
 2. Subsequences, Limsup, Liminf. (Ch 10 - 12 in Ross) \\ 2. Subsequences, Limsup, Liminf. (Ch 10 - 12 in Ross) \\
-3. Compactness and Topology 101 (Ch 13 in Ross, Chapter 2 in Rudin) \\+3. Topology (Ch 13 in Ross, Chapter 2 in Rudin) \\
 4. Series (Ch 14) \\ 4. Series (Ch 14) \\
 5. Continuity, Uniform Continuity, Uniform Convergence (Chapter 4, 7 in Rudin)\\ 5. Continuity, Uniform Continuity, Uniform Convergence (Chapter 4, 7 in Rudin)\\
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-1.1 Supremums and Infimums+==== Supremums and Infimums ====
  
 When you were probably 5 years old, you probably learned about what a maximum or minimum element of the set is. Such concepts seem clear cut; however, things get confusing when we introduce certain open sets (e.g. (-5, 5). When you were probably 5 years old, you probably learned about what a maximum or minimum element of the set is. Such concepts seem clear cut; however, things get confusing when we introduce certain open sets (e.g. (-5, 5).
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 More mathematical way of putting this (supremum is sup and infimum is inf): More mathematical way of putting this (supremum is sup and infimum is inf):
-$$\sup S = x \iff (\forall s \in S, s \leq x) \land (\forall y < x, \exists s_b \in S: y < s_b < x)$$ +$$\sup S = x \iff (\forall s \in S, s \leq x) \land (\forall y < x, \exists s_b \in S: y < s_b \leq x)$$ 
-$$\inf S = x \iff (\forall s \in S, s \geq x) \land (\forall y > x, \exists s_b \in S: y > s_b > x)$$+$$\inf S = x \iff (\forall s \in S, s \geq x) \land (\forall y > x, \exists s_b \in S: y > s_b \geq x)$$
  
 ** E.g.** ** E.g.**
  
- $$\sup \{x * y: x, y \in \{ 1, 2, 3, 4\}\} = 16$$ since $\{x * y: x, y \in \{ 1, 2, 3, 4\}\} = \{1, 2, 3, 4, 6, 8, 9, 12, 16\}$. All elements in the set are less than or equal to 16 and for any real number x less than 16, there exists a number in the set greater than x and less than or equal to 16; hence making 16 is an upper bound. Consider that for any z under 16 we always have an element in the set that is greater than z, e.g. 16. This makes 16 a ** minimum upper bound ** of the set.+ $$\sup \{x \cdot y: x, y \in \{ 1, 2, 3, 4\}\} = 16$$ since $\{x \cdot y: x, y \in \{ 1, 2, 3, 4\}\} = \{1, 2, 3, 4, 6, 8, 9, 12, 16\}$. All elements in the set are less than or equal to 16 and for any real number x less than 16, there exists a number in the set greater than x and less than or equal to 16; hence making 16 is an upper bound. Consider that for any z under 16 we always have an element in the set that is greater than z, e.g. 16. This makes 16 a ** minimum upper bound ** of the set.
  
  
-$$\inf \{x * y: x, y \in \{ 1, 2, 3, 4\}\} = 1$$. All elements in the set are greater than or equal to 1; hence making it an upper bound. Consider that for any z over 1 we always have an element in the set that is less than z, e.g. 1. This makes 1 a ** maximum lower bound ** of the set.+$$\inf \{x \cdot y: x, y \in \{ 1, 2, 3, 4\}\} = 1$$. All elements in the set are greater than or equal to 1; hence making it an upper bound. Consider that for any z over 1 we always have an element in the set that is less than z, e.g. 1. This makes 1 a ** maximum lower bound ** of the set.
  
 ** E.g. ** ** E.g. **
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 $\frac{q_1}{q_2}$ where $q_1, q_2$ are two integer factors of $c_0$ and $c_n$ respectively. $\frac{q_1}{q_2}$ where $q_1, q_2$ are two integer factors of $c_0$ and $c_n$ respectively.
  
-** 1.2 Sequences **+==== Sequences ==== 
 A sequence is an ordered set of numbers/terms. If a sequence $(x_n)$ in metric space $M$ converges to limit l, then we know that A sequence is an ordered set of numbers/terms. If a sequence $(x_n)$ in metric space $M$ converges to limit l, then we know that
 $$\forall \epsilon > 0, \exists N > 0: \forall n > N, d(x_n, l) < \epsilon$$. As a sidenote, d is a metric of metric space M. (We'll go over metric spaces in Section 2, but when dealing with $\mathbb{R}^n$ d is the Cartesian distance function in $\mathbb{R}^n$. $$\forall \epsilon > 0, \exists N > 0: \forall n > N, d(x_n, l) < \epsilon$$. As a sidenote, d is a metric of metric space M. (We'll go over metric spaces in Section 2, but when dealing with $\mathbb{R}^n$ d is the Cartesian distance function in $\mathbb{R}^n$.
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   - Division Rule (only if both numerator and denominator converge as $n \to \infty$.   - Division Rule (only if both numerator and denominator converge as $n \to \infty$.
  
-** Monotonically Increasing and Decreasing **+==== Monotonically Increasing and Decreasing ====
  
 Although it may be redundant, I thought that it would be best to establish the definitions of monotonically increasing and decreasing. This will come in handy in the next subsection. Although it may be redundant, I thought that it would be best to establish the definitions of monotonically increasing and decreasing. This will come in handy in the next subsection.
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-** Subsequences **+==== Subsequences ====
  
 We define a subsequence $s_{n_{k}}$ of $s_n$ as an ordered set of points in $s_n$ such that $\forall k \in \mathbb{N}, n_k < n_{k+1}$. We define a subsequence $s_{n_{k}}$ of $s_n$ as an ordered set of points in $s_n$ such that $\forall k \in \mathbb{N}, n_k < n_{k+1}$.
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-** 3.1 Metric Spaces **+==== Metric Spaces ====
 For most of our math lives up to Math 53, we have dealt with the cartesian space. We learned that distance between two points in 3d space is $d(x, y) = \sqrt{(x_0 - y_0)^2 + (x_1 - y_1)^2 + (x_2 - y_2)^2}$. Then in Math 54 (or in Physics 89) we met linear spaces in terms of vectors, functions, polynomials, and all the familiar constructs we could think of. Now, we use an ever more generalized idea of a space and distance. For most of our math lives up to Math 53, we have dealt with the cartesian space. We learned that distance between two points in 3d space is $d(x, y) = \sqrt{(x_0 - y_0)^2 + (x_1 - y_1)^2 + (x_2 - y_2)^2}$. Then in Math 54 (or in Physics 89) we met linear spaces in terms of vectors, functions, polynomials, and all the familiar constructs we could think of. Now, we use an ever more generalized idea of a space and distance.
  
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 Therefore $E' = \emptyset$. And therefore $E = \overline{E}$ and thus $[0, 0.5]$ is closed. This illustrates that just because $E$ is open does not necessarily mean that it is not closed. Therefore $E' = \emptyset$. And therefore $E = \overline{E}$ and thus $[0, 0.5]$ is closed. This illustrates that just because $E$ is open does not necessarily mean that it is not closed.
  
-** Other Definitions **+==== Useful results on Open Sets Closed Sets ==== 
 + 
 +  - A set is open iff its complement is closed. 
 +  - Given that $\bigcup S_i$ is a union of open sets, $\bigcup S_i$ is open. 
 +  - Given that $\bigcap S_i$ is the intersection of closed sets, $\bigcap S_i$ is closed. 
 +  - Given that $\bigcup S_i$ is a union of **finitely many** closed sets, $\bigcup S_i$ is closed. 
 +  - Given that $\bigcap S_i$ is the intersection of open sets, $\bigcap S_i$ is open. 
 + 
 +==== Induced Topology ==== 
 + 
 +Sometimes, we want to create our own metric space and transfer over properties. How these properties transfer over can be summarized by the following drawing (credits to Peng Zhou). 
 + 
 +{{ math104-s21:s:inducedtopology.jpg?400 |}} 
 + 
 +In this diagram there are two ways to create induced topology. We can either obtain the topology from an induced metric or from the old set itself. 
 + 
 +One key idea to note is that (and I quote from course notes) **Given $A \subseteq S$ and (S, d) is a metric space, we can equip A with an induced metric. $E \subseteq A$ is open iff $\exists$ open set $ E_2 \subseteq S$ such that $E = E_2 \cap S$. From how a complement of an open set in ambient space is closed, we can use this theorem to tell alot topologically about the set.** 
 + 
 + 
 +==== Other Basic Topology Definitions ====
   - A Set is ** countable ** iff there exists a one to one mapping between the elements of the set to the set of natural numbers.   - A Set is ** countable ** iff there exists a one to one mapping between the elements of the set to the set of natural numbers.
   - An ** open ball ** (or neighborhood as referred to in Rudin), about point x and with radius r is $B(r; x) = \{y \in M: d(y, x) < r\}$ where (M, d) is the ambient metric space of x.   - An ** open ball ** (or neighborhood as referred to in Rudin), about point x and with radius r is $B(r; x) = \{y \in M: d(y, x) < r\}$ where (M, d) is the ambient metric space of x.
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   - $S$ is bounded iff $\exists r: \forall p, q \in S, d(p, q) \leq r$.   - $S$ is bounded iff $\exists r: \forall p, q \in S, d(p, q) \leq r$.
   - $S$ is dense in $X$ iff every element in X is in $S$ or is a limit point of $S$.   - $S$ is dense in $X$ iff every element in X is in $S$ or is a limit point of $S$.
 +  - $S$ is a perfect set iff $S$ is closed and $S$ does not have any isolated points.
   - $diam(S)$, the diameter of $S$ is given to be $diam(S) = \sup_{x, y \in S} d(x, y)$.   - $diam(S)$, the diameter of $S$ is given to be $diam(S) = \sup_{x, y \in S} d(x, y)$.
  
 ** E.g.: ** We can say that in the set $\{\frac{1}{n}: n in \mathbb{N}\}$ has a limit point $0$ since we know that $\lim \frac{1}{n} = 0$ and therefore $\forall \epsilon > 0, \exists N \in \mathbb{N}: \forall n \geq N, d(\frac{1}{n}, 0) = |\frac{1}{n} - 0| < \epsilon$. Therefore, $\forall \epsilon > 0, B(\epsilon; x)$ ** E.g.: ** We can say that in the set $\{\frac{1}{n}: n in \mathbb{N}\}$ has a limit point $0$ since we know that $\lim \frac{1}{n} = 0$ and therefore $\forall \epsilon > 0, \exists N \in \mathbb{N}: \forall n \geq N, d(\frac{1}{n}, 0) = |\frac{1}{n} - 0| < \epsilon$. Therefore, $\forall \epsilon > 0, B(\epsilon; x)$
  
-** Useful results on Open Sets Closed Sets** 
-  - A set is open iff its complement is closed. 
-  - Given that $\bigcup S_i$ is a union of open sets, $\bigcup S_i$ is open. 
-  - Given that $\bigcap S_i$ is the intersection of closed sets, $\bigcap S_i$ is closed. 
-  - Given that $\bigcup S_i$ is a union of **finitely many** closed sets, $\bigcup S_i$ is closed. 
-  - Given that $\bigcap S_i$ is the intersection of open sets, $\bigcap S_i$ is open. 
  
  
-** Compactness **+==== Compactness ====
  
 In a very abstract way, we can think of compact as the mathematical way of saying controlled, contained or small; to get a gist of what compactness mean, one may take some of the interesting analogies/metaphors/layman concepts expressed in [[https://blogs.scientificamerican.com/roots-of-unity/what-does-compactness-really-mean/|this article]] from the //Scientific American.// In a very abstract way, we can think of compact as the mathematical way of saying controlled, contained or small; to get a gist of what compactness mean, one may take some of the interesting analogies/metaphors/layman concepts expressed in [[https://blogs.scientificamerican.com/roots-of-unity/what-does-compactness-really-mean/|this article]] from the //Scientific American.//
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   - Given that $\{C_{\alpha}\}$ is a collection of compact sets such that the intersection of any finite number of sets in the collection is nonempty, we know that $\bigcap C_{\alpha}$ is nonempty.   - Given that $\{C_{\alpha}\}$ is a collection of compact sets such that the intersection of any finite number of sets in the collection is nonempty, we know that $\bigcap C_{\alpha}$ is nonempty.
  
-** Connectedness **+==== Connectedness ====
  
 We know that $A$ are connected iff we cannot write $A$ as a union of two disjoint open sets. We know that $A$ are connected iff we cannot write $A$ as a union of two disjoint open sets.
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 Given real functions $f, g$ continuous over interval $[x_1, x_2]$ and differentiable over $(x_1, x_2)$. We know that $\exists x \in (x_1, x_2): (f(x_2) - f(x_1))g'(x) = (g(x_2) - g(x_1))(f'(x))$. Given real functions $f, g$ continuous over interval $[x_1, x_2]$ and differentiable over $(x_1, x_2)$. We know that $\exists x \in (x_1, x_2): (f(x_2) - f(x_1))g'(x) = (g(x_2) - g(x_1))(f'(x))$.
 +
 +=== Intermediate Value Theorems for Derivatives ===
 +
 +In order to make an analog to the intermediate value theorem for continuous functions, we use make the following statement: 
 +** Given that f is differentiable over $[a, b]$, $f'(a) < f'(b)$, and $\mu$ is in $(f'(a), f'(b))$, I know that $\exists c \in (a, b): f'(x_0) = \mu$.**
 +
 +One important thing to note here is that this ** does not result from the intermediate value theorem for continuous functions. ** Rather, it originates from the idea of global minima. The proof is very simple:
 +
 +Consider $g(x) = f(x) - \mu x$. We know $g'(a) = f'(a) - \mu < 0$ and $g'(b) = f'(a) - \mu > 0$. Now,since $[a, b]$ is compact, we know that there exists a global minimum for $f$. It cannot be $a, b$ since $f'(a) < 0, f'(b) > 0$ are nonzero. Therefore, we must say that the global minimum for $f$ must be $x_0 \in (a, b)$. Since $f$ is continuous on $[a, b]$ it must be the case that $x_0$ is a local minimum as well. Thus, $f'(x_0) = 0$. 
  
 ==== L'Hopital's Rule ==== ==== L'Hopital's Rule ====
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