MATH 521 (Analysis I) Definitions and Theorems

Fall 2022 (Lecture: Jordan Ellenberg, Note: Trevor Leslie)

Ruixuan Tu (ruixuan.tu@wisc.edu), University of Wisconsin-Madison

Walter Rudin: Principles of Mathematical Analysis
Trevor Leslie: Course Note for Math 521: Analysis I
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1. Def A set is a collection of objects called elements
2. Def $A$ is a subset of $B$ ($A\subset B$ or $A \subseteq B$) if $\forall x\in A, x\in B$
3. Def If $A$, $B$ are sets, the Cartesian product $A\times B$ is the set of ordered pairs $\forall a\in A, \forall b\in B, (a, b)$
4. Def $A$, $B$ are sets; a function $F: A\to B$ is a subset $F\subset A\times B$ s.t. $(a, b)\in F\land (a, b')\in F\implies b=b'$; $\forall a\in A, \exists b\in B, (a, b)\in F$
5. Def $f: A\to B$ is injective if $\forall a, a'\in A, f(a)=f(a')\implies a=a'$, i.e., if $\forall b\in B$ at most 1 $a\in A$ with $f(a)=b$
6. Def $f: A\to B$ is surjective if $\forall b\in B, \exists a\in A, F(a)=b$, i.e., if $\forall b\in B$ at least 1 $a\in A$ with $f(a)=b$
7. Def $f: A\to B$ is bijective iff $f$ is both injective and surjective
8. Fact When $f: A\to B$ is bijective, $\exists f^{-1}: B\to A$ s.t. $f^{-1}(b)=\text{the unique }a\in A\text{ s.t. }f(a)=b$
9. Def Given $f: A\to B, g: B\to C$; the function composition is $gf$ or $g\circ f: A\to C$ s.t. $\forall a\in A, g\circ f(a)=g(f(a))$
10. Fact Target of $f$ must match source of $g$ that not all pairs of functions can be composed
11. Fact Function composition is not commutative
12. Def A relation from $A$ to $B$ is a subset $R\subset A\times B$
13. Def An ordering on $S$ is a relation $\leq$ on $S\times S$ s.t. (comparability) $\forall x, y\in S$ exactly one of $x<y$, $y<x$, $x=y$ is true; (transitivity) $\forall x, y, z\in S$ if $x\leq y$ and $y\leq z$, then $x\leq z$
14. Def An ordered set is a set $(S, \leq)$ with an order $\leq$ on $S$
15. Fact If $x\leq y$ and $y\leq x$, then $x=y$
16. Def $S$ is an ordered set, $E\subset S$; if $\exists s\in S$ s.t. $\forall e\in E, s\geq e$, then $E$ is bounded above in $S$, and $s$ is an upper bound for $E$
17. Def $s$ is a least upper bound, $\sup E$, or the supremum of $E$ if $s$ is an upper bound for $E$; $t$ is an upper bound for $E\implies t\geq s$
18. Def Similarly, greatest lower bound for E is called infimum or $\inf E$
19. Def A field is a set where we can do arithmetic ($+$, $-$, $\times$, $\div$ except 0). We require the existence of $0$, $1$ and the rules $0+x=x$, $x+y=y+x$, $(x+y)+z=x+(y+z)$, $x(y+z)=xy+xz$, $xy=yx$
20. Def An ordered field is a field $F$ with an ordering $<$ s.t. for $x, y, z\in F$ with $y<z$, $x+y<x+z$; if $x,y\in F$, $x>0, y>0$, $xy>0$
21. Fact In an ordered field, $\forall x\in F$ $x^2\geq 0$
22. Fact A bounded ordered set $e\subset S$ has at most one supremum
23. Fact $\left\{ x\in \mathbb{Q}_{>0}: x^2<2 \right\}$ has no supremum
24. Def A nonempty set $S$ has the least upper bound property if every nonempty $E\subset S$ which is bounded above has a least upper bound
25. Thm If $S$ has the least upper bound property, then $S$ has the greatest lower bound property
26. Def An ordered field is complete if it has the least upper bound property
27. Rmk $\mathbb{Q}$ is not complete, a finite ordered set has the least upper bound property
28. Thm There is a complete ordered field $\mathbb{R}$
29. Def If $F, G$ are ordered fields, an isomorphism is a function $f: F\to G$ s.t. $f$ is a bijection; $f(x)>f(y)$ iff $x>y$; $f(x+y)=f(x)+f(y)$; $f(xy)=f(x)f(y)$ ($F, G$ being "the same field with the elements labelled differently")
30. Prop $F=\mathbb{R}$ has the Archimedean property: for any $x, y>0$, $\exists n\in \mathbb{Z}_{>0}$ s.t. $nx>y$
31. Prop $F=\mathbb{R}, n\in F_{>0}$, then there is a unique $y\in F$ with $y^n=x$ ("$n^\text{th}$ roots exist")
32. Lemma $a$ is an upper bound for $\left\{ z: z\geq 0, z^n\leq x \right\}$ iff $a^n\geq x$
33. Def A Cauchy sequence of $\mathbb{Q}$ is a sequence $\mathbf{a}=a_1, a_2, a_3, \dots$ s.t. $\forall \epsilon>0, \exists N\in \mathbb{Z}_{>0}$ s.t. $\forall i,j>N, |a_i-a_j|<\epsilon$
34. Def $\mathbb{Q}$ is dense in $\mathbb{R}$ if: if $x, y\in \mathbb{R}$ and $x\lt y$, then $\exists p\in \mathbb{Q}$ s.t. $x\lt p\lt y$
35. Fact Irrational numbers $\mathbb{R}\setminus \mathbb{Q}$ is dense in $\mathbb{R}$
36. Def An equivalence relation on $S$ is a relation $\sim\in S\times S$ s.t. $\forall x, x\sim x$; $\forall x, y, x\sim y \iff y\sim x$; $\forall x, y, z, x\sim y \land y\sim z\implies x\sim z$ (which is a bijection)
37. Def An equation relation $\sim$ partitions $S$ into disjoint subsets (i.e. equivalence classes) $s_i$, each one of the form $\left\{ s\in S: s\sim s_0 \right\}$ for some $s_0$
38. Def A real number is an equivalence class of Cauchy sequences under this relation
39. Prop If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are Cauchy sequences, $\mathbf{a}\sim \mathbf{b}$ and $\mathbf{b}\sim \mathbf{c}$, then $\mathbf{a}\sim \mathbf{c}$
40. Def $\mathbf{a}>\mathbf{b}$ if $\exists \delta>0, N$ s.t. $a_i>b_i+\delta$ for all $i>N$
41. Prop If $x,y\in \mathbb{R}$, exactly one of $x<y, x>y, x=y$ is true
42. Def A Cauchy sequence of $\mathbb{R}$ is a sequence $\mathbf{x}=x_1,x_2,\dots$ $x_i\in \mathbb{R}$ s.t. $\forall \epsilon>0,\exists N$ s.t. $|x_i-y|<\epsilon$
43. Def A sequence of reals $\mathbf{x}$ has limit $y$ if $\forall \epsilon>0, \exists N$ s.t. $\forall i>N, |x_i-y|<\epsilon$
44. Thm $\mathbb{R}$ has the least upper bound property
45. Thm (Monotone Convergence Thm) A non-decreasing sequence of $x_i\in \mathbb{R}$ which is bounded above is Cauchy and thus converges to a limit (everything that rises must converge)
46. Def $\overline{R}$, the extended reals, is an ordered set $\mathbb{R} \cup \left\{ -\infty, \infty \right\}$
47. Def (Limit) If $\mathbf{x}=x_1,x_2,\dots$ is a sequence in $\overline{R}$, $\lim \mathbf{x}=\infty$ if $\forall E, \exists N, \forall i > N, x_i>E$
48. Fact $\overline{R}$ is not a field
49. Def The equivalence classes under $\sim$ are called cardinalities
50. Fact The equivalence classes of finite sets are $\mathbb{N}$ (by pigeonhole principle)
51. Def A set $S$ is infinite if $\exists f: S\hookrightarrow S$ (monomorphism) an injection which is not a bijection
52. Fact An infinite set cannot be equivalent to a finite set
53. Def $S$ is countably infinite if $S\sim \mathbb{Z}_{>0}$
54. Prop $\mathbb{Z}$ is countable
55. Fact
    1. If $S, T$ are countable, so is $S\cup T$
    2. If $S$ countable, any subset of $S$ is countably infinite or finite
    3. If $S, T$ are countable, so is $S\times T$
56. Def if $S, T$ are sets, we denote by $S^T$ the set of functions from $T$ to $S$
57. Fact
    1. If $S, T$ are finite sets, $|S^T|=|S|^{|T|}$
    2. $\left\{ 0, 1 \right\} ^T=$ set of subsets of $T$
    3. $S^\emptyset=\emptyset$
58. Thm (Cantor’s Diagonal Argument) There are uncountable sets, i.e., $\left\{ 0,1 \right\} ^{\mathbb{Z}_{\geq 0}}$
59. Cor $\mathbb{R}$ is uncountable
60. Def A vector space over a field $k$ (a $k$-vector space) is a set $V$, whose elements are called vectors with operations $+$ (addition) and $\cdot$ (scalar multiplication)
61. Def $V$ is a vector space over $\mathbb{R}$; a norm on $V$ is a function $||\cdot||: V\to \mathbb{R}$ s.t. (nonnegativity) $\forall v\in V, ||v||\geq 0,$ and $||v||=0$ iff $v=\mathbf{0}$; (homogeneity) $||\lambda v||=|\lambda|||v|| \forall \lambda\in \R, v\in V$; (triangle inequality) $\forall v, w\in V, ||v+w||\leq ||v||+||w||$
62. Rmk the set of functions $f:\R\to \R$ bounded between $-1$ and $1$ are not a vector space
63. Def The norm ball $B_{v, ||\cdot||}(a)$ is the set $\left\{ v\in V: ||v||\leq a \right\}$
64. Def A subset $S$ of a vector space $V$ is convex if $\forall v, w\in S$, the line segment $\overline{vw}$ is continued in $S$
65. Fact Norm balls are always convex
66. Def
    1. ($L^p$ norm) $||(x_1,\dots,x_n)||_p=(|x_1|^p+\cdots+|x_n|^p)^{\frac{1}{p}}$
    2. ($L^\infty$ norm) $||(x_1,\dots,x_n)||_\infty=\max_i |x_i|$
    3. ($\sup$ norm)$||f||_{\sup}=\sup_{x}|f(x)|$
67. Fact $||v||_1\geq ||v||_2\geq ||v||_\infty$, $||v||_1\leq 2||v||_\infty$, $||v||_1\leq \sqrt{d}||v||_2$ for $v\in \R^d$
68. Def An inner product space is a vector space $V$ with a function $<, >: V\times V\to \R$ s.t. (symmetry) $\left<x,y \right> =\left<y, x \right>$; $\left<x_1+x_2, y \right> =\left<x_1, y \right>+\left<x_2, y \right>$; (linearity in first component) $\left< \lambda x+y, z\right> =\lambda \left<x, z \right> + \left<y, z \right>$; (positive-definiteness) $\left<x,x \right>\geq 0$ and $\left<x, x \right> =0$ iff $x=0$
69. Thm For every inner product space $V$, $||v||\coloneqq \left<v,v \right>^{\frac{1}{2}}$ is a norm, i.e., every inner product space is a normed space
70. Thm (Cauchy-Schwarz Inequality) $\left<x,y \right>\leq ||x|| ||y||$
71. Cor If $v,w\in \R^n$, Cauchy-Schwarz says $\cos \theta=\frac{\left<v,w \right>}{||v||||w||}\leq 1$, where $\theta$ is the angle between $v$ and $w$
72. Fact correlation coefficient $\rho$ measures the relation between the variables, i.e., the angle between $\vec{x}$ and $\vec{y}$ if everything is normalized to the mean $0$
    1. $\rho=1$ iff $\forall i, a>0, y_i=a x_i$ iff $\theta=0$
    2. $\rho>0$ iff $\vec{x}, \vec{y}$ are at an acute angle; $\rho<0$ for obtuse angle; $\rho=0$ for orthogonal/perpendicular
73. Def A metric space is a set $X$ with function $d: X\times X\to \R$ s.t. (nonnegativity) $d(x,y)\geq 0$, with $d(x,y)=0$ iff $x=y$; (symmetry) $d(x,y)=d(y,x)$; (triangle inequality) $d(x,z)\leq d(x,y)+d(y,z)$
74. Fact If $V$ is a normed vector space, then the function $d(x, y)=||x-y||$ is a metric
75. Def $X$ is metric space with metric $d$; the open ball of radius $r$ around $x\in X$ is $B_{X, d}(x, r)=\left\{ y\in X: d(x,y)<r \right\}$
76. Def $X$ is a metric space, $S\subset X$; the interior of $S$ with respect to $X$, denoted $\text{int}_X(S)$, is the set $\left\{ s\in S: \exists \epsilon>0, B_X(s, \epsilon)\subset S \right\}$
77. Def A subset $S\subset X$ is open if $\text{int}_X(S)=S$
78. Thm $\forall S\subset X$, if $\text{int}(\text{int}(S))=\text{int}(S)$, then $\text{int}(S)$ is open
79. Prop (Properties of open/closed sets)
    1. Any union of an arbitrary collection of open sets is open
    2. Any intersection of a finite collection of open sets is open
    3. Any union of a finite collection of closed sets is closed
    4. Any intersection of an arbitrary collection of closed sets is closed
80. Prop Every open set $S\subset X$ is a union of some collection of open balls
81. Def $X$ is a metric space, $p\in X$; a neighborhood of $p$ is an open set $U\ni p$
82. Def $E$ is a subset of metric space $X$; $x\in X$ is an accumulation point or limit point of $E$ in $X$ (i.e., $x\in L_X(E)$) if for every neighborhood $U\ni x$, $U\cap E$ contains a point not equal to $x$, i.e., $\forall B(x, \epsilon), \epsilon>0$
83. Def A point $x$ is an isolated point of $E$ in $X$ if $x\in E$ and $x$ is not a limit point of $E$ in $X$
84. Prop If $X=\R^n$ with Euclidean metric and $E\subset X$, then any point in $\text{int}(E)$ is an accumulation point for $E$
85. Def Let $x_1,x_2,\dots$ be a sequence of points in a metric space $X$; $\lim_{i\to \infty} x_i=x$ if, for every neighborhood $U$ of $x$, $\exists N_U$ s.t. $\forall i>N_U$, $x_i\in U$; i.e.; $\forall \epsilon>0, \exists N_{\epsilon}$ s.t. $\forall i>N_{\epsilon}$, $x_i\in B(x,\epsilon)$ i.e. $d(x, x_i)<\epsilon$
86. Thm Let $E\subset X$; $x\in L_X(E)$ iff $\exists$ a sequence $x_1,x_2,\dots\in E\setminus \{x\}$ with $\lim_{i\to \infty} x_i=x$
87. Def $E$ is closed in $X$ if $L_X(E)\subset E$
88. Def $E\subset X$ is closed if its complement $\overline{E}$ ($\coloneqq X\setminus E$) is open
89. Def The closure of $E$ in $X$ is $\text{clos}(E)=E\cup L_X(E)$; i.e.; $\text{clos}(E)=\overline{\text{int}(\overline{E})}$, or $\text{clos}(E)=$ intersection of all closed subsets of $X$ containing $E$
90. Rmk $\text{clos}(E)=E$ iff $E$ is closed
91. Def $x$ is a subsequential limit of $x_1,x_2,\dots$ if $\exists$ a sequence $x_{i_1}, x_{i_2}, \dots$ ($i_1<i_2<\cdots$) with $\lim_{j\to \infty}x_{i_j}=x$
92. Prop If $\lim_{i\to\infty} x_i=x$, then $x$ is the only subsequential limit
93. Thm (Bolzano–Weierstrass Thm) Any bounded sequence of real numbers has a subsequential limit
94. Thm (Baby Bolzano–Weierstrass Thm) If $K$ is a finite set of real numbers, then any sequence $x_1,x_2,\dots\in K$ has a subsequential limit (by pigeonhole principle)
95. Prop Let $x_1,x_2,\dots$ a sequence in $X$, $E=\left\{ x_1,x_2,\dots \right\}\subset X$; if $x$ is an accumulation point of $E$, it is a subsequential limit of $x_1,x_2,\dots$
96. Def Let $K$ be a subset of a metric space $X$; $K$ is compact if, for any collection of open sets of $X$ ($\left\{ U_s \right\}_{s\in S}$) which covers $K$ (i.e. $K\subset \cup_{s\in S}U_s$), there is a finite subcollection $U_1, U_2, \dots, U_N$ which still covers $K$
97. Fact A finite set $K$ is compact
98. Fact Any unbounded subset of $\R$ (including $\R, \Z$) is noncompact
99. Fact $(0,1)$ is noncompact; $[0, 1]$ is compact
100. Thm (Heine-Borel Thm) $\forall S\subset \R^n$, $S$ is closed and bounded iff $S$ is compact
101. Thm Let $x_1,x_2,\dots$ be an infinite sequence in a compact subset $K\subset X$; then $x_1, x_2,\dots$ has a convergent subsequence where limit is in $K$
102. Thm An infinite subset $S$ of a compact set $K\subset X$ has an accumulation point in $K$
103. Cor If $x_1,x_2,\dots$ is a sequence of points in compact set $K$, then $x_1,x_2,\dots$ has a subsequential limit in $K$
104. Fact Compact sets are always closed so all accumulation points/subsequential limits are in $K$
105. Prop A closed subset $Y$ of a compact set $K\subset X$ is compact
106. Def $X$ is a metric space, $E\subset X$. $E$ is bounded if $\exists B_X(x,r)\supset E$
107. Def Let $X$ be a metric space, $E\subset X$. $E$ is dense in $X$ if $\text{clos}_X(E) = X$
108. Prop If $K\subset X$ is compact, then $K$ is bounded
109. Def An n-cell/box in $\R^n$ is a set $\left\{ a_1\leq x_1\leq b_1, \dots, a_n\leq x_n\leq b_n \right\}=[a_1, b_1]\times \cdots \times [a_n, b_n]$
110. Prop (Nested Interval Theorem) If $[a_1, b_1]\supset [a_2, b_2]\supset [a_3, b_3]\supset \cdots$ is a sequence of nested closed intervals in $\R$, $\exists x\in \R$ s.t. $\forall i, x\in [a_i, b_i]$
111. Def A sequence $x_1, x_2, \dots$ in a metric space $X$ is a Cauchy sequence if $\forall \epsilon>0, \exists N_{\epsilon}$ s.t. $\forall i, j>N_{\epsilon}, d(x_i, x_j)<\epsilon$
112. Prop If $x_1, x_2, \dots$ is a Cauchy sequence, it has at most one subsequential limit, $\lim_i x_i=x$
113. Def If every Cauchy sequence in $X$ converges, $X$ is complete
114. Prop Every compact metric space is complete
115. Def The Cantor set $E\coloneqq \cap_{i=0}^{\infty} E_i$ with $E_0=[0, 1], E_1=[0, \frac{1}{3}]\cup [\frac{2}{3}, 1], E_2=[0, \frac{1}{9}]\cup [\frac{2}{9}, \frac{1}{3}]\cup [\frac{2}{3}, \frac{7}{9}]\cup [\frac{8}{9}, 1], \dots$
116. Fact The Cantor set $E$ is an intersection of closed sets, so it is closed, but (1) $E$ has no isolated points; (2) $E$ contains no closed interval
117. Def (Roughly Minkowski Dimension) Let $X$ be a bounded metric space. Define $\forall \epsilon>0, C(x, \epsilon)=$ smallest number of $\epsilon$-balls that can cover $X$. The dimension of $X$ is the inverse exponent of $C(x, \epsilon)$
118. Fact $C(x, \epsilon)$ for: (1) line segment $\sim c\cdot \epsilon^{-1}$, (2) square $\sim c\cdot \epsilon^{-2}$, (3) three points $=3\cdot \epsilon^{0}$, (4) line circumscribing square $\sim 4c \epsilon^{-1}$, (5) circumference of circle $\sim c'\cdot \epsilon^{-1}$
119. Prop The Cantor set is uncountable, closed, and contains no positive length interval
120. Def (Limit) Let $X$ be a metric space, $E\subset X$ be a subset, $f: E\to Y$, and $a$ an accumulation point in $L_X(E)$, $b\in Y$. $\lim_{x\to a} f(x)=b$, if for every neighborhood $U$ of $b$, $\exists$ a neighborhood $V$ of $a$ s.t. $\forall x\in V\setminus a, f(x)\in U$. Equivalently, $\forall \epsilon\gt 0, \exists \delta$ s.t. if $0\lt d(x,a)\lt \delta$ then $d(f(x), b)\lt \epsilon$
121. Rmk In $\mathbb{R}$, $0\lt |x-a|\lt \delta, |f(x)-b|\lt \epsilon$
122. Def (Notion) For $f: X\to Y$, $S\subset X$. $f(S)=\left\{ f(x): x\in S \right\}$
123. Def $X, Y$ are metric spaces, $f: X\to Y$ is a function. $f$ is continuous at a point $p\in X$ if $\forall$ neighborhood $V$ of $f(p)$, $\exists$ a neighborhood $U$ of $p$ s.t. $f(U)\subset V$
124. Prop The following are equivalent: (1) $f$ is continuous of $p$; (2) $\lim_{x\to p}f(x)=f(p)$
125. Def $f: X\to Y$ is continuous if:
     1. (Analysts’ Definition) it is continuous at every point $p$ of $X$, i.e., $\lim_{x\to p} f(x)=f(p)$
     2. (Topologists’ Definition, Equivalent) $\forall$ open subset $V\subset Y$, $f^{-1}(V)=\left\{ x\in X: f(x)\in V \right\}$ is open in $X$
126. Fact about continuity
     1. Constant functions are continuous
     2. The identity map $i: X\to X$ is continuous
     3. If $f, s: X\to \mathbb{R}$ are continuous, so are $f+g$, $f-g$, $fg$ (though not necessarily $\frac{f}{g}$)
     4. Any polynomial $p: \mathbb{R} \to \mathbb{R}$ is continuous
     5. If $f: X\to Y$ and $g: Y\to Z$ are continuous, so is $g\circ f: X\to Z$
     6. If $E\subset Y$, and $g: Y\to Z$ is continuous, then $g|_E: E\to Z$ is continuous ($g|_E$: $g$ is restricted to $E$, i.e., $\forall e\in E, g|_E(e)=g(e)$)
     7. A function $X\to \mathbb{R}^n$ $(f_1, f_2, \dots, f_n)$ is continuous iff $\forall i$, $f_i$ is constant
127. Thm (Weak Extreme Value Theorem) A function on a finite set has a maximum
128. Thm (Extreme Value Theorem) Let $K$ be a compact metric space and $f: K\to \mathbb{R}$ be a continuous function. Then $\exists x\in K$ s.t. $f(x)=\sup_{p\in K}f(p)$
129. Def A function whose input is another function is functional (e.g., $X$ be space of $\left[ 0, 1 \right]\to \mathbb{R}$, $F:X\to \mathbb{R}$ functional)
130. Def A metric space $X$ is connected if:
     1. (Analysts’ Definition) $\not\exists$ surjective continuous function $f: X\to \left\{ 0, 1 \right\}$ (Note $\left\{ 0 \right\}$ is open)
     2. (Topologists’ Definition, Equivalent) $\not\exists$ two nonempty open subsets $U_0, U_1\subset X$ s.t. $U_0\cap U_1=\emptyset$ and $U_0\cup U_1=X$ (In this case, $U_1=U_0^c$ is closed, from where also clopen)
131. Fact about connectedness: (1) The Cantor set is totally disconnected: the only connected subsets are single points; yet the Cantor set has no isolated points. (2) $\mathbb{Q}$ is not connected
132. Def Let $f_1,f_2,\dots$ be functions $X\to Y$, and $f(x)=\lim_{n\to \infty} f_n(x)$ when this limit exists. If $\forall x\in X$ the limit exists, we say $f_i\to f$ pointwise
133. Fact about sequences of functions
     1. For bump function $f_1(x)$ a bump, $f_2(x)=f_1(x-1)$, $f_3(x)=f_1(x-2)$, $\dots$; pointwise limit $f(n)=\lim_{n\to \infty}f_n(n)=0$. Note that $f_i\to 0$ is not true in the $\sup$ norm
     2. The pointwise limit of continuous functions is not always continuous
134. Def Let $f_1, f_2, \dots$ be functions $X\to \mathbb{R}$. We say $f_1, f_2, \dots$ converges pointwise to $f$ if, $\forall x\in X$, $\lim_{i\to \infty} f_i(x)=f(x)$, i.e., $\forall x\in X$, $\forall \epsilon>0$, $\exists N_{x,\epsilon}$ s.t. $\forall n\gt N_{x, \epsilon}$, $|f_n(x)-f(x)|\lt \epsilon$
135. Def We say $f_1, f_2, \dots$ converges uniformly to $f$ if, $\forall \epsilon\gt 0$, $\exists N_{\epsilon}$ s.t. $\forall x\in X$, $\forall n\gt N_{\epsilon}$, $|f_n(x)-f(x)|\lt \epsilon$
136. Rmk on uniform convergence
     1. If $f_1,f_2,\dots$ converges uniformly to $f$, it converges pointwise to $f$
     2. If $f_1, f_2, \dots$ are bounded functions in $X$, then $f_1,f_2, \dots\to f$ uniformly iff $\lim_{i\to \infty} f_i=f$ if the metric space of bounded functions with $\sup$ norm
     3. $X=[0, c], c\lt 1$, $f_n(x)=x^n$ does uniformly converge to $0$
137. Def We say a function $f:[a,b]\to \mathbb{R}$ is $L^2$-null if $\int_a^b |f(x)|^2 dx=0$
138. Def We say $f$ and $g$ are equivalent in $L^2$ ($f\sim_{L^2} g$) if $f-g$ is $L^2$-null
139. Def The space of $L^2$ functions in $[a,b]$ is the set of convergence classes for $\sim_{L^2}$ (equivalence class)
140. Fact Denote $f\sim g$ if $||f-g||_2=0$. The set of equivalence classes for this relation forms a normed vector space called the space of $L^2$ functions
141. Thm Let $X$ be a set, $Y$ be a complete metric space. $B(X)$ is the normed vector space of bounded functions $X\to Y$. Then $B(X)$ is complete
142. Def A complete normed vector space is called a Banach space
143. Def A complete inner product space is called a Hilbert space
144. Thm (Uniform Limit Theorem) Let $f_1,f_2, \dots$ be a subsequence of continuous functions uniformly converging to $f$. Then $f$ is continuous
145. Rmk about Uniform Limit Theorem
     1. As a consequence, if $B^c(X)\subset B(X)$ is the subspace of continuous functions, then $B^c$ is closed and complete
     2. As a consequence, if $B^{cts}(X)\subset \text{bounded }B(X)$ is the subspace of bounded continuous functions, then $B^{cts}$ is closed and complete. $f_1, f_2, \dots$ is a Cauchy sequence of $cts$ (continuous) bounded functions. $f_1, f_2, \dots$ uniformly converges to some $f$ in $B(X)$ because $B(X)$ is complete. By Uniform Limit Theorem, $f$ is continuous, $f\in B^{cts}(X)$
146. Def Let $f: (a, b)\to \mathbb{R}$. We write $f(x+)=q$ with $x\in (a, b)$ to mean: $\lim_{n\to \infty} f(x_n)=q$ for every sequence $x_1, x_2, \dots \subset (x, b)$, i.e., "limit as $f$ goes to $x$ from above". $f(x-)=q$ means same thing but for sequences in $(a, x)$
147. Prop $f$ is continuous at $x$ iff $f(x+)$ exists, $f(x-)$ exists, and $f(x-)=f(x)=f(x+)$. In particular, this applies to sequences used to define $f(x+)$ and $f(x-)$
148. Lemma If a subsequence can be partitioned into a finite union of subsequences, each converging to $L$, then the sequences converge to $L$
149. Def Types of discontinuities
     1. removable: $f(x-)=f(x+)$ but $f(x)$ does not equal to these
     2. jump: $f(x-)$ and $f(x+)$ both exist but are not equal
     3. essential: either $f(x-)$ or $f(x+)$ does not exist
150. Thm (Froda’s Theorem) Let $f:\mathbb{R} \to \mathbb{R}$ be a function, $\text{Discont}(f)\subset \mathbb{R}$ be the set $\left\{ x\in \mathbb{R}: f \text{ is discontinnuous at } x \right\}$. A set $S\subset \mathbb{R}$ can be $\text{Discont}(f)$ for some $f: \mathbb{R}\to \mathbb{R}$ iff it is the union of countably many closed sets (i.e., any closed set)
151. Def A function $f: (a,b)\to \mathbb{R}$ is monotone nondecreasing if $f(y)\geq f(x)$ whenever $y\geq x$ and strictly increasing if $f(y)\gt f(x)$ whenever $y\gt x$
152. Prop If $f$ is monotone nondecreasing, then $f(x-)=\sup_{y\lt x} f(y)$ and $f(x+)=\inf_{y\gt x} f(y)$, i.e., $f(x-)\leq f(x)$ and $f(x)\leq f(x+)$. All discontinuities are jump discontinuities
153. Thm If $f$ is monotone nondecreasing, $\text{Discont}(f)$ is countable
154. Rmk (Intuition of Uniform Convergence) $f_n$ is contained in $\epsilon$-band around $f$ $\iff$ $\forall x, f_n(x)\in (f(x)-\epsilon, f(x)+\epsilon)$ $\iff$ $\forall x, |f_n(x)-f(x)|\lt \epsilon$
155. Def Let $s_1, s_2, \dots$ be a sequence of real numbers. We define $\mathop{\lim \sup}_{n\to \infty} s_n=\lim_{n\to \infty} \sup_{k\geq n} s_k$
156. Def Let $s_1, s_2, \dots$ be a sequence of real numbers. We define $\mathop{\lim \inf}_{n\to \infty} s_n=\lim_{n\to \infty} \inf_{k\geq n} s_k$
157. Prop $\mathop{\lim \sup}$ and $\mathop{\lim \inf}$ always exist in $\mathbb{R} \cup \left\{ \infty, -\infty \right\}$, and exist in $\mathbb{R}$ iff the sequence is bounded
158. Rmk $\lim s_n=\infty$ is not the same thing as "$s_n$ unbounded above"
159. Prop $\mathop{\lim \sup} s_n=\infty \iff s_n \text{ unbounded above}$. $\mathop{\lim \inf} s_n=-\infty \iff s_n \text{ unbounded below}$. If $s_n$ is unbounded, so is every tail of $s_n$, so $\sup_{k\geq n}s_k=\infty$ for every $n$
160. Prop $\mathop{\lim \sup}$ is the supremum of all subsequential limits. $\mathop{\lim \inf}$ is the infimum of all subsequential limits
     1. If $s_n\to s=\lim s_n$, then $\mathop{\lim \sup} s_n=\mathop{\lim \inf} s_n=\lim s_n$
     2. Conversely, if $\mathop{\lim \sup} s_n=\mathop{\lim \inf} s_n$, then $\lim s_n$ exists and equals to these
161. Def We define partial sums $s_n=\sum_{i=1}^{n} a_i$
162. Def Define $\sum_{i=1}^{\infty} a_i=\lim_{n\to \infty}\sum_{i=1}^{n} a_i$ be a series with a sequence of partial sums
163. Def If the limit $\lim_{n\to \infty}\sum_{i=1}^{n} a_i$ exists, in which case we say, the infinite series $\sum_{i=1}^{\infty} a_i$ converges
164. Rmk For this definition, make sure we are summing something with an ordering (e.g., no sum of all rational numbers)
165. Rmk For $n>m$, $s_n-s_m=\sum_{i=1}^{n} a_i-\sum_{i=1}^{m} a_i=\sum_{i=m+1}^{n} a_i$. If $\sum a_i$ converges, $\lim a_i$ exists and $=0$ (but not conversely)
166. Example (Geometric series)
     1. $x\in \mathbb{R}$, $\sum_{i=0}^{\infty} x^i=1+x+x^2+x^3+\cdots$. If $|x|\geq 0$, $x^i$ does not converge to 0, so the series does not converge; when $|x|\lt 1$, this converges to $\frac{1}{1-x}$
     2. $s_n=\sum_{i=1}^{n} x^n=1+x+\cdots+x^n=\frac{1-x^{n+1}}{1-x}$
167. Example (Harmonic series) $\sum_{n=1}^{\infty} \frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots=1+\underset{= \frac{2}{4}}{ \left( \frac{1}{2}+\frac{1}{3} \right) }+\underset{\geq \frac{4}{8}}{ \left( \frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7} \right) }+\underset{\geq \frac{8}{16}}{\left( \frac{1}{8}+\cdots \right)}$
168. Prop (Comparison test) Suppose $\sum c_n$ be convergent series, $\sum d_n$ be divergent series. Let $\sum a_n$ be some series we wish to analyze. If $\exists N$ s.t. $\forall n\gt N$, $|a_n|\leq c_n$, then $\sum a_n$ converges. If $\exists N$ s.t. $\forall n\gt N$, $|a_n|\geq d_n$, then $\sum a_n$ diverges.
169. Prop (Tail test) $\sum_{i=1}^{\infty} a_i$ converges iff all its tails $\sum_{i=m}^{\infty} a_i$ converge
170. Prop (Cauchy condensation test) Suppose $a_n$ is non-negative and monotone non-increasing. Then $\sum_{n=1}^{\infty} a_n$ converges iff $\sum_{k=0}^{\infty} 2^k a_{2^k}$ converges
171. Prop $\sum_{n=1}^{\infty} \frac{1}{n^s}$ converges iff $s>1$
172. Prop (Root test) If $\mathop{\lim \sup}_{n\to \infty}\sqrt[n]{|a_n|}\lt 1$, then $\sum_{k=1}^{\infty} a_k$ converges. If $\mathop{\lim \sup}_{n\to \infty}\sqrt[n]{|a_n|}\gt 1$, then $\sum_{k=1}^{\infty} a_k$ diverges
173. Prop (Ratio test) If $\mathop{\lim \sup}_{n\to \infty} \left| \frac{a_{n+1}}{a_{n}} \right|\lt 1$, $\sum a_n$ converges. If $\mathop{\lim \inf}_{n\to \infty} \left| \frac{a_{n+1}}{a_{n}} \right|\gt 1$, $\sum a_n$ diverges
174. Prop (Alternating series test) Let $a_n$ be a sequence with alternating signs and suppose $|a_n|$ are monotone decreasing. Then $\sum a_n$ converges iff $\lim a_n=0$
175. Thm Suppose $a_1, a_2, \dots$ is a sequence s.t. $\sum_{i=1}^{\infty} |a_i|$ converges (absolute convergent), then $\sum a_i$ converges, and if $a_i'$ is any arrangement of $a_i$, then $\sum a_i'$ converges to $\sum a_i$. If $\sum a_i$ is convergent but not absolute convergent, I can find a rearrangement which diverges and a rearrangement which sums to $x$ $\forall x\in \mathbb{R}$
176. Thm (Weierstrass M-test) Let $M_1, M_2, \dots$ a sequence of non-negative reals s.t. $\sum_{i=1}^{\infty} M_i$ converges, and $f_1, f_2, \dots$ a sequence of functions $X\to \mathbb{R}$ s.t. $\forall x\in X, |f_i(x)|\leq M_i$. Then $\sum_{i=0}^{\infty} f_i$ converges uniformly
177. Def Some functions can be expressed as infinite sums of trigonometric functions, i.e., Fourier series. There is Gibbs phenomenon, the oscillatory behavior around a jump discontinuity
178. Def We say $f$ is differentiable of $x\in X$ if for all sequences $t_1, t_2, \dots \to x$, $\lim_{t\to x}\frac{f(t)-f(x)}{t-x}$ exists. In this case, the limit is always the same and we call it $f'(x)$
179. Rmk If the limit exists, $\lim_{t\to x} f(t)-f(x)=0$, so $f$ differentiable at $x$ $\implies$ $f$ continuous at $x$, but not $\impliedby$ (e.g., corner)
180. Prop Let $f:(a, b)\to \mathbb{R}$, let $x$ be an maximum, i.e., $\forall y\in (a, b), f(x)\geq f(y)$, and suppose $f$ is differentiable at $x$. Then $f'(x)=0$
181. Lemma If $f$ is continuous on $[a, b]$ and $f(a)=f(b)$, then $f$ has an extremum in $(a, b)$
182. Thm (Mean Value Theorem) Let $f: [a,b]\to \mathbb{R}$ be continuous, differentiable on $(a, b)$. $\exists x^{*}\in (a, b)$ s.t. $f'(x^{*})=\frac{f(b)-f(a)}{b-a}$ (Note $h(x)=\frac{(f(b)-f(a))x-(b-a)f(x)}{t}$)
183. Def A partition of $[a, b]$ is a "finite set of real numbers": $a=x_0\leq x_1\leq \cdots \leq x_n=b$, $\Delta x_i=x_{i+1}-x_i$
184. Def Riemann upper integral of $f$ on $[a, b]$ is $\inf_P U(P, f)$, lower integral of $f$ on $[a, b]$ is $\sup_P L(P, F)$
185. Prop $L(P_1, f)\leq U(P_2, f)$ for every pair of partitions $P_1, P_2$
186. Def We say $P^*$ is a refinement of $P$ if every breakpoint of $P$ is a breakpoint of $P^*$
187. Lemma If $P^*$ a refinement of $P$, then $L(P, f)\leq L(P^*, f)\leq U(P^*, f)\leq U(P, f)$
188. Fact If $P_1$ and $P_2$ are partitions, there is a partition $Q$ which is a common refinement of both (simply take the union of $\text{breakpoints}(P_1)\cup \text{breakpoints}(P_2)$), $L(P_1, f)\leq L(Q, f)\leq U(Q, f)\leq U(P_2, f)$
189. Def If lower integral = upper integral ($\inf_P U(P, f)=\sup_P L(P, f)$), we say $f$ is Riemann integrable on $[a, b]$ and define $\int_a^b f(x) ~dx$ to be the common value
190. Rmk (Dirichlet function) $\chi_{\mathbb {Q}}(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}$ with $\chi_{\mathbb{Q}}: [a, b]\to \{0, 1\}$, is non-integrable ($U(P, \chi_{\mathbb{Q}})=b-a, L(P, \chi_{\mathbb{Q}})=0$)
191. Rmk $\chi_{\text{Cantor set}}(x)={\begin{cases}1&x\in \text{Cantor set} \\0&x\notin \text{Cantor set} \end{cases}}$ with $\chi_{\text{Cantor set}}: [0, 1]\to \{0, 1\}$, is integrable with $\int_0^1 \chi_{\text{Cantor set}}(x) ~dx=0$ ($U(P_k, f)=(\frac{2}{3})^k+\epsilon, \inf_P U(P, f)\leq \inf_k U(P_k, f)=0$)
192. Thm If $f$ is continuous, then $f$ is integrable
193. Thm If $f$ is (bounded) monotone, then $f$ is integrable
194. Fact
     1. If $c\in [a, b]$ and $f$ integrable on $[a, b]$ then $\int_a^b f(x) ~dx-\int_a^c f(x) ~dx=\int_c^b f(x) ~dx$
     2. If $f$ is integrable, then $|f|$ is integrable, and $\int_a^b |f(x)| ~dx\geq |\int_a^b f(x) ~dx|$ (triangle inequality)
195. Def (Indefinite integral) Let $f$ integrable on $[a, b]$, then we define $F: [a, b]\to \mathbb{R}$ by $F(x)=\int_a^x f(t) ~dt$
196. Thm $F$ is continuous and Lipschitz
197. Thm (Fundamental Theorem of Calculus, FToC) If $f$ is continuous at $x_0\in [a, b]$ and integrable, then $F$ is differentiable at $x_0$, and $F'(x_0)=f(x_0)$