MATH 521 (Analysis I) Definitions and Theorems

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

Ruixuan Tu (, 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 AA is a subset of BB (ABA\subset B or ABA \subseteq B) if xA,xB\forall x\in A, x\in B
  3. Def If AA, BB are sets, the Cartesian product A×BA\times B is the set of ordered pairs aA,bB,(a,b)\forall a\in A, \forall b\in B, (a, b)
  4. Def AA, BB are sets; a function F:ABF: A\to B is a subset FA×BF\subset A\times B s.t. (a,b)F(a,b)F    b=b(a, b)\in F\land (a, b')\in F\implies b=b'; aA,bB,(a,b)F\forall a\in A, \exists b\in B, (a, b)\in F
  5. Def f:ABf: A\to B is injective if a,aA,f(a)=f(a)    a=a\forall a, a'\in A, f(a)=f(a')\implies a=a', i.e., if bB\forall b\in B at most 1 aAa\in A with f(a)=bf(a)=b
  6. Def f:ABf: A\to B is surjective if bB,aA,F(a)=b\forall b\in B, \exists a\in A, F(a)=b, i.e., if bB\forall b\in B at least 1 aAa\in A with f(a)=bf(a)=b
  7. Def f:ABf: A\to B is bijective iff ff is both injective and surjective
  8. Fact When f:ABf: A\to B is bijective, f1:BA\exists f^{-1}: B\to A s.t. f1(b)=the unique aA s.t. f(a)=bf^{-1}(b)=\text{the unique }a\in A\text{ s.t. }f(a)=b
  9. Def Given f:AB,g:BCf: A\to B, g: B\to C; the function composition is gfgf or gf:ACg\circ f: A\to C s.t. aA,gf(a)=g(f(a))\forall a\in A, g\circ f(a)=g(f(a))
  10. Fact Target of ff must match source of gg that not all pairs of functions can be composed
  11. Fact Function composition is not commutative
  12. Def A relation from AA to BB is a subset RA×BR\subset A\times B
  13. Def An ordering on SS is a relation \leq on S×SS\times S s.t. (comparability) x,yS\forall x, y\in S exactly one of x<yx<y, y<xy<x, x=yx=y is true; (transitivity) x,y,zS\forall x, y, z\in S if xyx\leq y and yzy\leq z, then xzx\leq z
  14. Def An ordered set is a set (S,)(S, \leq) with an order \leq on SS
  15. Fact If xyx\leq y and yxy\leq x, then x=yx=y
  16. Def SS is an ordered set, ESE\subset S; if sS\exists s\in S s.t. eE,se\forall e\in E, s\geq e, then EE is bounded above in SS, and ss is an upper bound for EE
  17. Def ss is a least upper bound, supE\sup E, or the supremum of EE if ss is an upper bound for EE; tt is an upper bound for E    tsE\implies t\geq s
  18. Def Similarly, greatest lower bound for E is called infimum or infE\inf E
  19. Def A field is a set where we can do arithmetic (++, -, ×\times, ÷\div except 0). We require the existence of 00, 11 and the rules 0+x=x0+x=x, x+y=y+xx+y=y+x, (x+y)+z=x+(y+z)(x+y)+z=x+(y+z), x(y+z)=xy+xzx(y+z)=xy+xz, xy=yxxy=yx
  20. Def An ordered field is a field FF with an ordering << s.t. for x,y,zFx, y, z\in F with y<zy<z, x+y<x+zx+y<x+z; if x,yFx,y\in F, x>0,y>0x>0, y>0, xy>0xy>0
  21. Fact In an ordered field, xF\forall x\in F x20x^2\geq 0
  22. Fact A bounded ordered set eSe\subset S has at most one supremum
  23. Fact {xQ>0:x2<2}\left\{ x\in \mathbb{Q}_{>0}: x^2<2 \right\} has no supremum
  24. Def A nonempty set SS has the least upper bound property if every nonempty ESE\subset S which is bounded above has a least upper bound
  25. Thm If SS has the least upper bound property, then SS has the greatest lower bound property
  26. Def An ordered field is complete if it has the least upper bound property
  27. Rmk Q\mathbb{Q} is not complete, a finite ordered set has the least upper bound property
  28. Thm There is a complete ordered field R\mathbb{R}
  29. Def If F,GF, G are ordered fields, an isomorphism is a function f:FGf: F\to G s.t. ff is a bijection; f(x)>f(y)f(x)>f(y) iff x>yx>y; f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y); f(xy)=f(x)f(y)f(xy)=f(x)f(y) (F,GF, G being "the same field with the elements labelled differently")
  30. Prop F=RF=\mathbb{R} has the Archimedean property: for any x,y>0x, y>0, nZ>0\exists n\in \mathbb{Z}_{>0} s.t. nx>ynx>y
  31. Prop F=R,nF>0F=\mathbb{R}, n\in F_{>0}, then there is a unique yFy\in F with yn=xy^n=x ("nthn^\text{th} roots exist")
  32. Lemma aa is an upper bound for {z:z0,znx}\left\{ z: z\geq 0, z^n\leq x \right\} iff anxa^n\geq x
  33. Def A Cauchy sequence of Q\mathbb{Q} is a sequence a=a1,a2,a3,\mathbf{a}=a_1, a_2, a_3, \dots s.t. ϵ>0,NZ>0\forall \epsilon>0, \exists N\in \mathbb{Z}_{>0} s.t. i,j>N,aiaj<ϵ\forall i,j>N, |a_i-a_j|<\epsilon
  34. Def Q\mathbb{Q} is dense in R\mathbb{R} if: if x,yRx, y\in \mathbb{R} and x<yx\lt y, then pQ\exists p\in \mathbb{Q} s.t. x<p<yx\lt p\lt y
  35. Fact Irrational numbers RQ\mathbb{R}\setminus \mathbb{Q} is dense in R\mathbb{R}
  36. Def An equivalence relation on SS is a relation S×S\sim\in S\times S s.t. x,xx\forall x, x\sim x; x,y,xy    yx\forall x, y, x\sim y \iff y\sim x; x,y,z,xyyz    xz\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 SS into disjoint subsets (i.e. equivalence classes) sis_i, each one of the form {sS:ss0}\left\{ s\in S: s\sim s_0 \right\} for some s0s_0
  38. Def A real number is an equivalence class of Cauchy sequences under this relation
  39. Prop If a,b,c\mathbf{a}, \mathbf{b}, \mathbf{c} are Cauchy sequences, ab\mathbf{a}\sim \mathbf{b} and bc\mathbf{b}\sim \mathbf{c}, then ac\mathbf{a}\sim \mathbf{c}
  40. Def a>b\mathbf{a}>\mathbf{b} if δ>0,N\exists \delta>0, N s.t. ai>bi+δa_i>b_i+\delta for all i>Ni>N
  41. Prop If x,yRx,y\in \mathbb{R}, exactly one of x<y,x>y,x=yx<y, x>y, x=y is true
  42. Def A Cauchy sequence of R\mathbb{R} is a sequence x=x1,x2,\mathbf{x}=x_1,x_2,\dots xiRx_i\in \mathbb{R} s.t. ϵ>0,N\forall \epsilon>0,\exists N s.t. xiy<ϵ|x_i-y|<\epsilon
  43. Def A sequence of reals x\mathbf{x} has limit yy if ϵ>0,N\forall \epsilon>0, \exists N s.t. i>N,xiy<ϵ\forall i>N, |x_i-y|<\epsilon
  44. Thm R\mathbb{R} has the least upper bound property
  45. Thm (Monotone Convergence Thm) A non-decreasing sequence of xiRx_i\in \mathbb{R} which is bounded above is Cauchy and thus converges to a limit (everything that rises must converge)
  46. Def R\overline{R}, the extended reals, is an ordered set R{,}\mathbb{R} \cup \left\{ -\infty, \infty \right\}
  47. Def (Limit) If x=x1,x2,\mathbf{x}=x_1,x_2,\dots is a sequence in R\overline{R}, limx=\lim \mathbf{x}=\infty if E,N,i>N,xi>E\forall E, \exists N, \forall i > N, x_i>E
  48. Fact R\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 N\mathbb{N} (by pigeonhole principle)
  51. Def A set SS is infinite if f:SS\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 SS is countably infinite if SZ>0S\sim \mathbb{Z}_{>0}
  54. Prop Z\mathbb{Z} is countable
  55. Fact
    1. If S,TS, T are countable, so is STS\cup T
    2. If SS countable, any subset of SS is countably infinite or finite
    3. If S,TS, T are countable, so is S×TS\times T
  56. Def if S,TS, T are sets, we denote by STS^T the set of functions from TT to SS
  57. Fact
    1. If S,TS, T are finite sets, ST=ST|S^T|=|S|^{|T|}
    2. {0,1}T=\left\{ 0, 1 \right\} ^T= set of subsets of TT
    3. S=S^\emptyset=\emptyset
  58. Thm (Cantor’s Diagonal Argument) There are uncountable sets, i.e., {0,1}Z0\left\{ 0,1 \right\} ^{\mathbb{Z}_{\geq 0}}
  59. Cor R\mathbb{R} is uncountable
  60. Def A vector space over a field kk (a kk-vector space) is a set VV, whose elements are called vectors with operations ++ (addition) and \cdot (scalar multiplication)
  61. Def VV is a vector space over R\mathbb{R}; a norm on VV is a function :VR||\cdot||: V\to \mathbb{R} s.t. (nonnegativity) vV,v0,\forall v\in V, ||v||\geq 0, and v=0||v||=0 iff v=0v=\mathbf{0}; (homogeneity) λv=λvλR,vV||\lambda v||=|\lambda|||v|| \forall \lambda\in \R, v\in V; (triangle inequality) v,wV,v+wv+w\forall v, w\in V, ||v+w||\leq ||v||+||w||
  62. Rmk the set of functions f:RRf:\R\to \R bounded between 1-1 and 11 are not a vector space
  63. Def The norm ball Bv,(a)B_{v, ||\cdot||}(a) is the set {vV:va}\left\{ v\in V: ||v||\leq a \right\}
  64. Def A subset SS of a vector space VV is convex if v,wS\forall v, w\in S, the line segment vw\overline{vw} is continued in SS
  65. Fact Norm balls are always convex
  66. Def
    1. (LpL^p norm) (x1,,xn)p=(x1p++xnp)1p||(x_1,\dots,x_n)||_p=(|x_1|^p+\cdots+|x_n|^p)^{\frac{1}{p}}
    2. (LL^\infty norm) (x1,,xn)=maxixi||(x_1,\dots,x_n)||_\infty=\max_i |x_i|
    3. (sup\sup norm)fsup=supxf(x)||f||_{\sup}=\sup_{x}|f(x)|
  67. Fact v1v2v||v||_1\geq ||v||_2\geq ||v||_\infty, v12v||v||_1\leq 2||v||_\infty, v1dv2||v||_1\leq \sqrt{d}||v||_2 for vRdv\in \R^d
  68. Def An inner product space is a vector space VV with a function <,>:V×VR<, >: V\times V\to \R s.t. (symmetry) <x,y>=<y,x>\left<x,y \right> =\left<y, x \right>; <x1+x2,y>=<x1,y>+<x2,y>\left<x_1+x_2, y \right> =\left<x_1, y \right>+\left<x_2, y \right>; (linearity in first component) <λx+y,z>=λ<x,z>+<y,z>\left< \lambda x+y, z\right> =\lambda \left<x, z \right> + \left<y, z \right>; (positive-definiteness) <x,x>0\left<x,x \right>\geq 0 and <x,x>=0\left<x, x \right> =0 iff x=0x=0
  69. Thm For every inner product space VV, v<v,v>12||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) <x,y>xy\left<x,y \right>\leq ||x|| ||y||
  71. Cor If v,wRnv,w\in \R^n, Cauchy-Schwarz says cosθ=<v,w>vw1\cos \theta=\frac{\left<v,w \right>}{||v||||w||}\leq 1, where θ\theta is the angle between vv and ww
  72. Fact correlation coefficient ρ\rho measures the relation between the variables, i.e., the angle between x\vec{x} and y\vec{y} if everything is normalized to the mean 00
    1. ρ=1\rho=1 iff i,a>0,yi=axi\forall i, a>0, y_i=a x_i iff θ=0\theta=0
    2. ρ>0\rho>0 iff x,y\vec{x}, \vec{y} are at an acute angle; ρ<0\rho<0 for obtuse angle; ρ=0\rho=0 for orthogonal/perpendicular
  73. Def A metric space is a set XX with function d:X×XRd: X\times X\to \R s.t. (nonnegativity) d(x,y)0d(x,y)\geq 0, with d(x,y)=0d(x,y)=0 iff x=yx=y; (symmetry) d(x,y)=d(y,x)d(x,y)=d(y,x); (triangle inequality) d(x,z)d(x,y)+d(y,z)d(x,z)\leq d(x,y)+d(y,z)
  74. Fact If VV is a normed vector space, then the function d(x,y)=xyd(x, y)=||x-y|| is a metric
  75. Def XX is metric space with metric dd; the open ball of radius rr around xXx\in X is BX,d(x,r)={yX:d(x,y)<r}B_{X, d}(x, r)=\left\{ y\in X: d(x,y)<r \right\}
  76. Def XX is a metric space, SXS\subset X; the interior of SS with respect to XX, denoted intX(S)\text{int}_X(S), is the set {sS:ϵ>0,BX(s,ϵ)S}\left\{ s\in S: \exists \epsilon>0, B_X(s, \epsilon)\subset S \right\}
  77. Def A subset SXS\subset X is open if intX(S)=S\text{int}_X(S)=S
  78. Thm SX\forall S\subset X, if int(int(S))=int(S)\text{int}(\text{int}(S))=\text{int}(S), then int(S)\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 SXS\subset X is a union of some collection of open balls
  81. Def XX is a metric space, pXp\in X; a neighborhood of pp is an open set UpU\ni p
  82. Def EE is a subset of metric space XX; xXx\in X is an accumulation point or limit point of EE in XX (i.e., xLX(E)x\in L_X(E)) if for every neighborhood UxU\ni x, UEU\cap E contains a point not equal to xx, i.e., B(x,ϵ),ϵ>0\forall B(x, \epsilon), \epsilon>0
  83. Def A point xx is an isolated point of EE in XX if xEx\in E and xx is not a limit point of EE in XX
  84. Prop If X=RnX=\R^n with Euclidean metric and EXE\subset X, then any point in int(E)\text{int}(E) is an accumulation point for EE
  85. Def Let x1,x2,x_1,x_2,\dots be a sequence of points in a metric space XX; limixi=x\lim_{i\to \infty} x_i=x if, for every neighborhood UU of xx, NU\exists N_U s.t. i>NU\forall i>N_U, xiUx_i\in U; i.e.; ϵ>0,Nϵ\forall \epsilon>0, \exists N_{\epsilon} s.t. i>Nϵ\forall i>N_{\epsilon}, xiB(x,ϵ)x_i\in B(x,\epsilon) i.e. d(x,xi)<ϵd(x, x_i)<\epsilon
  86. Thm Let EXE\subset X; xLX(E)x\in L_X(E) iff \exists a sequence x1,x2,E{x}x_1,x_2,\dots\in E\setminus \{x\} with limixi=x\lim_{i\to \infty} x_i=x
  87. Def EE is closed in XX if LX(E)EL_X(E)\subset E
  88. Def EXE\subset X is closed if its complement E\overline{E} (XE\coloneqq X\setminus E) is open
  89. Def The closure of EE in XX is clos(E)=ELX(E)\text{clos}(E)=E\cup L_X(E); i.e.; clos(E)=int(E)\text{clos}(E)=\overline{\text{int}(\overline{E})}, or clos(E)=\text{clos}(E)= intersection of all closed subsets of XX containing EE
  90. Rmk clos(E)=E\text{clos}(E)=E iff EE is closed
  91. Def xx is a subsequential limit of x1,x2,x_1,x_2,\dots if \exists a sequence xi1,xi2,x_{i_1}, x_{i_2}, \dots (i1<i2<i_1<i_2<\cdots) with limjxij=x\lim_{j\to \infty}x_{i_j}=x
  92. Prop If limixi=x\lim_{i\to\infty} x_i=x, then xx 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 KK is a finite set of real numbers, then any sequence x1,x2,Kx_1,x_2,\dots\in K has a subsequential limit (by pigeonhole principle)
  95. Prop Let x1,x2,x_1,x_2,\dots a sequence in XX, E={x1,x2,}XE=\left\{ x_1,x_2,\dots \right\}\subset X; if xx is an accumulation point of EE, it is a subsequential limit of x1,x2,x_1,x_2,\dots
  96. Def Let KK be a subset of a metric space XX; KK is compact if, for any collection of open sets of XX ({Us}sS\left\{ U_s \right\}_{s\in S}) which covers KK (i.e. KsSUsK\subset \cup_{s\in S}U_s), there is a finite subcollection U1,U2,,UNU_1, U_2, \dots, U_N which still covers KK
  97. Fact A finite set KK is compact
  98. Fact Any unbounded subset of R\R (including R,Z\R, \Z) is noncompact
  99. Fact (0,1)(0,1) is noncompact; [0,1][0, 1] is compact
  100. Thm (Heine-Borel Thm) SRn\forall S\subset \R^n, SS is closed and bounded iff SS is compact
  101. Thm Let x1,x2,x_1,x_2,\dots be an infinite sequence in a compact subset KXK\subset X; then x1,x2,x_1, x_2,\dots has a convergent subsequence where limit is in KK
  102. Thm An infinite subset SS of a compact set KXK\subset X has an accumulation point in KK
  103. Cor If x1,x2,x_1,x_2,\dots is a sequence of points in compact set KK, then x1,x2,x_1,x_2,\dots has a subsequential limit in KK
  104. Fact Compact sets are always closed so all accumulation points/subsequential limits are in KK
  105. Prop A closed subset YY of a compact set KXK\subset X is compact
  106. Def XX is a metric space, EXE\subset X. EE is bounded if BX(x,r)E\exists B_X(x,r)\supset E
  107. Def Let XX be a metric space, EXE\subset X. EE is dense in XX if closX(E)=X\text{clos}_X(E) = X
  108. Prop If KXK\subset X is compact, then KK is bounded
  109. Def An n-cell/box in Rn\R^n is a set {a1x1b1,,anxnbn}=[a1,b1]××[an,bn]\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 [a1,b1][a2,b2][a3,b3][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\R, xR\exists x\in \R s.t. i,x[ai,bi]\forall i, x\in [a_i, b_i]
  111. Def A sequence x1,x2,x_1, x_2, \dots in a metric space XX is a Cauchy sequence if ϵ>0,Nϵ\forall \epsilon>0, \exists N_{\epsilon} s.t. i,j>Nϵ,d(xi,xj)<ϵ\forall i, j>N_{\epsilon}, d(x_i, x_j)<\epsilon
  112. Prop If x1,x2,x_1, x_2, \dots is a Cauchy sequence, it has at most one subsequential limit, limixi=x\lim_i x_i=x
  113. Def If every Cauchy sequence in XX converges, XX is complete
  114. Prop Every compact metric space is complete
  115. Def The Cantor set Ei=0EiE\coloneqq \cap_{i=0}^{\infty} E_i with E0=[0,1],E1=[0,13][23,1],E2=[0,19][29,13][23,79][89,1],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 EE is an intersection of closed sets, so it is closed, but (1) EE has no isolated points; (2) EE contains no closed interval
  117. Def (Roughly Minkowski Dimension) Let XX be a bounded metric space. Define ϵ>0,C(x,ϵ)=\forall \epsilon>0, C(x, \epsilon)= smallest number of ϵ\epsilon-balls that can cover XX. The dimension of XX is the inverse exponent of C(x,ϵ)C(x, \epsilon)
  118. Fact C(x,ϵ)C(x, \epsilon) for: (1) line segment cϵ1\sim c\cdot \epsilon^{-1}, (2) square cϵ2\sim c\cdot \epsilon^{-2}, (3) three points =3ϵ0=3\cdot \epsilon^{0}, (4) line circumscribing square 4cϵ1\sim 4c \epsilon^{-1}, (5) circumference of circle cϵ1\sim c'\cdot \epsilon^{-1}
  119. Prop The Cantor set is uncountable, closed, and contains no positive length interval
  120. Def (Limit) Let XX be a metric space, EXE\subset X be a subset, f:EYf: E\to Y, and aa an accumulation point in LX(E)L_X(E), bYb\in Y. limxaf(x)=b\lim_{x\to a} f(x)=b, if for every neighborhood UU of bb, \exists a neighborhood VV of aa s.t. xVa,f(x)U\forall x\in V\setminus a, f(x)\in U. Equivalently, ϵ>0,δ\forall \epsilon\gt 0, \exists \delta s.t. if 0<d(x,a)<δ0\lt d(x,a)\lt \delta then d(f(x),b)<ϵd(f(x), b)\lt \epsilon
  121. Rmk In R\mathbb{R}, 0<xa<δ,f(x)b<ϵ0\lt |x-a|\lt \delta, |f(x)-b|\lt \epsilon
  122. Def (Notion) For f:XYf: X\to Y, SXS\subset X. f(S)={f(x):xS}f(S)=\left\{ f(x): x\in S \right\}
  123. Def X,YX, Y are metric spaces, f:XYf: X\to Y is a function. ff is continuous at a point pXp\in X if \forall neighborhood VV of f(p)f(p), \exists a neighborhood UU of pp s.t. f(U)Vf(U)\subset V
  124. Prop The following are equivalent: (1) ff is continuous of pp; (2) limxpf(x)=f(p)\lim_{x\to p}f(x)=f(p)
  125. Def f:XYf: X\to Y is continuous if:
    1. (Analysts’ Definition) it is continuous at every point pp of XX, i.e., limxpf(x)=f(p)\lim_{x\to p} f(x)=f(p)
    2. (Topologists’ Definition, Equivalent) \forall open subset VYV\subset Y, f1(V)={xX:f(x)V}f^{-1}(V)=\left\{ x\in X: f(x)\in V \right\} is open in XX
  126. Fact about continuity
    1. Constant functions are continuous
    2. The identity map i:XXi: X\to X is continuous
    3. If f,s:XRf, s: X\to \mathbb{R} are continuous, so are f+gf+g, fgf-g, fgfg (though not necessarily fg\frac{f}{g})
    4. Any polynomial p:RRp: \mathbb{R} \to \mathbb{R} is continuous
    5. If f:XYf: X\to Y and g:YZg: Y\to Z are continuous, so is gf:XZg\circ f: X\to Z
    6. If EYE\subset Y, and g:YZg: Y\to Z is continuous, then gE:EZg|_E: E\to Z is continuous (gEg|_E: gg is restricted to EE, i.e., eE,gE(e)=g(e)\forall e\in E, g|_E(e)=g(e))
    7. A function XRnX\to \mathbb{R}^n (f1,f2,,fn)(f_1, f_2, \dots, f_n) is continuous iff i\forall i, fif_i is constant
  127. Thm (Weak Extreme Value Theorem) A function on a finite set has a maximum
  128. Thm (Extreme Value Theorem) Let KK be a compact metric space and f:KRf: K\to \mathbb{R} be a continuous function. Then xK\exists x\in K s.t. f(x)=suppKf(p)f(x)=\sup_{p\in K}f(p)
  129. Def A function whose input is another function is functional (e.g., XX be space of [0,1]R\left[ 0, 1 \right]\to \mathbb{R}, F:XRF:X\to \mathbb{R} functional)
  130. Def A metric space XX is connected if:
    1. (Analysts’ Definition) ∄\not\exists surjective continuous function f:X{0,1}f: X\to \left\{ 0, 1 \right\} (Note {0}\left\{ 0 \right\} is open)
    2. (Topologists’ Definition, Equivalent) ∄\not\exists two nonempty open subsets U0,U1XU_0, U_1\subset X s.t. U0U1=U_0\cap U_1=\emptyset and U0U1=XU_0\cup U_1=X (In this case, U1=U0cU_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) Q\mathbb{Q} is not connected
  132. Def Let f1,f2,f_1,f_2,\dots be functions XYX\to Y, and f(x)=limnfn(x)f(x)=\lim_{n\to \infty} f_n(x) when this limit exists. If xX\forall x\in X the limit exists, we say fiff_i\to f pointwise
  133. Fact about sequences of functions
    1. For bump function f1(x)f_1(x) a bump, f2(x)=f1(x1)f_2(x)=f_1(x-1), f3(x)=f1(x2)f_3(x)=f_1(x-2), \dots; pointwise limit f(n)=limnfn(n)=0f(n)=\lim_{n\to \infty}f_n(n)=0. Note that fi0f_i\to 0 is not true in the sup\sup norm
    2. The pointwise limit of continuous functions is not always continuous
  134. Def Let f1,f2,f_1, f_2, \dots be functions XRX\to \mathbb{R}. We say f1,f2,f_1, f_2, \dots converges pointwise to ff if, xX\forall x\in X, limifi(x)=f(x)\lim_{i\to \infty} f_i(x)=f(x), i.e., xX\forall x\in X, ϵ>0\forall \epsilon>0, Nx,ϵ\exists N_{x,\epsilon} s.t. n>Nx,ϵ\forall n\gt N_{x, \epsilon}, fn(x)f(x)<ϵ|f_n(x)-f(x)|\lt \epsilon
  135. Def We say f1,f2,f_1, f_2, \dots converges uniformly to ff if, ϵ>0\forall \epsilon\gt 0, Nϵ\exists N_{\epsilon} s.t. xX\forall x\in X, n>Nϵ\forall n\gt N_{\epsilon}, fn(x)f(x)<ϵ|f_n(x)-f(x)|\lt \epsilon
  136. Rmk on uniform convergence
    1. If f1,f2,f_1,f_2,\dots converges uniformly to ff, it converges pointwise to ff
    2. If f1,f2,f_1, f_2, \dots are bounded functions in XX, then f1,f2,ff_1,f_2, \dots\to f uniformly iff limifi=f\lim_{i\to \infty} f_i=f if the metric space of bounded functions with sup\sup norm
    3. X=[0,c],c<1X=[0, c], c\lt 1, fn(x)=xnf_n(x)=x^n does uniformly converge to 00
  137. Def We say a function f:[a,b]Rf:[a,b]\to \mathbb{R} is L2L^2-null if abf(x)2dx=0\int_a^b |f(x)|^2 dx=0
  138. Def We say ff and gg are equivalent in L2L^2 (fL2gf\sim_{L^2} g) if fgf-g is L2L^2-null
  139. Def The space of L2L^2 functions in [a,b][a,b] is the set of convergence classes for L2\sim_{L^2} (equivalence class)
  140. Fact Denote fgf\sim g if fg2=0||f-g||_2=0. The set of equivalence classes for this relation forms a normed vector space called the space of L2L^2 functions
  141. Thm Let XX be a set, YY be a complete metric space. B(X)B(X) is the normed vector space of bounded functions XYX\to Y. Then B(X)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 f1,f2,f_1,f_2, \dots be a subsequence of continuous functions uniformly converging to ff. Then ff is continuous
  145. Rmk about Uniform Limit Theorem
    1. As a consequence, if Bc(X)B(X)B^c(X)\subset B(X) is the subspace of continuous functions, then BcB^c is closed and complete
    2. As a consequence, if Bcts(X)bounded B(X)B^{cts}(X)\subset \text{bounded }B(X) is the subspace of bounded continuous functions, then BctsB^{cts} is closed and complete. f1,f2,f_1, f_2, \dots is a Cauchy sequence of ctscts (continuous) bounded functions. f1,f2,f_1, f_2, \dots uniformly converges to some ff in B(X)B(X) because B(X)B(X) is complete. By Uniform Limit Theorem, ff is continuous, fBcts(X)f\in B^{cts}(X)
  146. Def Let f:(a,b)Rf: (a, b)\to \mathbb{R}. We write f(x+)=qf(x+)=q with x(a,b)x\in (a, b) to mean: limnf(xn)=q\lim_{n\to \infty} f(x_n)=q for every sequence x1,x2,(x,b)x_1, x_2, \dots \subset (x, b), i.e., "limit as ff goes to xx from above". f(x)=qf(x-)=q means same thing but for sequences in (a,x)(a, x)
  147. Prop ff is continuous at xx iff f(x+)f(x+) exists, f(x)f(x-) exists, and f(x)=f(x)=f(x+)f(x-)=f(x)=f(x+). In particular, this applies to sequences used to define f(x+)f(x+) and f(x)f(x-)
  148. Lemma If a subsequence can be partitioned into a finite union of subsequences, each converging to LL, then the sequences converge to LL
  149. Def Types of discontinuities
    1. removable: f(x)=f(x+)f(x-)=f(x+) but f(x)f(x) does not equal to these
    2. jump: f(x)f(x-) and f(x+)f(x+) both exist but are not equal
    3. essential: either f(x)f(x-) or f(x+)f(x+) does not exist
  150. Thm (Froda’s Theorem) Let f:RRf:\mathbb{R} \to \mathbb{R} be a function, Discont(f)R\text{Discont}(f)\subset \mathbb{R} be the set {xR:f is discontinnuous at x}\left\{ x\in \mathbb{R}: f \text{ is discontinnuous at } x \right\}. A set SRS\subset \mathbb{R} can be Discont(f)\text{Discont}(f) for some f:RRf: \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)Rf: (a,b)\to \mathbb{R} is monotone nondecreasing if f(y)f(x)f(y)\geq f(x) whenever yxy\geq x and strictly increasing if f(y)>f(x)f(y)\gt f(x) whenever y>xy\gt x
  152. Prop If ff is monotone nondecreasing, then f(x)=supy<xf(y)f(x-)=\sup_{y\lt x} f(y) and f(x+)=infy>xf(y)f(x+)=\inf_{y\gt x} f(y), i.e., f(x)f(x)f(x-)\leq f(x) and f(x)f(x+)f(x)\leq f(x+). All discontinuities are jump discontinuities
  153. Thm If ff is monotone nondecreasing, Discont(f)\text{Discont}(f) is countable
  154. Rmk (Intuition of Uniform Convergence) fnf_n is contained in ϵ\epsilon-band around ff     \iff x,fn(x)(f(x)ϵ,f(x)+ϵ)\forall x, f_n(x)\in (f(x)-\epsilon, f(x)+\epsilon)     \iff x,fn(x)f(x)<ϵ\forall x, |f_n(x)-f(x)|\lt \epsilon
  155. Def Let s1,s2,s_1, s_2, \dots be a sequence of real numbers. We define limsupnsn=limnsupknsk\mathop{\lim \sup}_{n\to \infty} s_n=\lim_{n\to \infty} \sup_{k\geq n} s_k
  156. Def Let s1,s2,s_1, s_2, \dots be a sequence of real numbers. We define liminfnsn=limninfknsk\mathop{\lim \inf}_{n\to \infty} s_n=\lim_{n\to \infty} \inf_{k\geq n} s_k
  157. Prop limsup\mathop{\lim \sup} and liminf\mathop{\lim \inf} always exist in R{,}\mathbb{R} \cup \left\{ \infty, -\infty \right\}, and exist in R\mathbb{R} iff the sequence is bounded
  158. Rmk limsn=\lim s_n=\infty is not the same thing as "sns_n unbounded above"
  159. Prop limsupsn=    sn unbounded above\mathop{\lim \sup} s_n=\infty \iff s_n \text{ unbounded above}. liminfsn=    sn unbounded below\mathop{\lim \inf} s_n=-\infty \iff s_n \text{ unbounded below}. If sns_n is unbounded, so is every tail of sns_n, so supknsk=\sup_{k\geq n}s_k=\infty for every nn
  160. Prop limsup\mathop{\lim \sup} is the supremum of all subsequential limits. liminf\mathop{\lim \inf} is the infimum of all subsequential limits
    1. If sns=limsns_n\to s=\lim s_n, then limsupsn=liminfsn=limsn\mathop{\lim \sup} s_n=\mathop{\lim \inf} s_n=\lim s_n
    2. Conversely, if limsupsn=liminfsn\mathop{\lim \sup} s_n=\mathop{\lim \inf} s_n, then limsn\lim s_n exists and equals to these
  161. Def We define partial sums sn=i=1nais_n=\sum_{i=1}^{n} a_i
  162. Def Define i=1ai=limni=1nai\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 limni=1nai\lim_{n\to \infty}\sum_{i=1}^{n} a_i exists, in which case we say, the infinite series i=1ai\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>mn>m, snsm=i=1naii=1mai=i=m+1nais_n-s_m=\sum_{i=1}^{n} a_i-\sum_{i=1}^{m} a_i=\sum_{i=m+1}^{n} a_i. If ai\sum a_i converges, limai\lim a_i exists and =0=0 (but not conversely)
  166. Example (Geometric series)
    1. xRx\in \mathbb{R}, i=0xi=1+x+x2+x3+\sum_{i=0}^{\infty} x^i=1+x+x^2+x^3+\cdots. If x0|x|\geq 0, xix^i does not converge to 0, so the series does not converge; when x<1|x|\lt 1, this converges to 11x\frac{1}{1-x}
    2. sn=i=1nxn=1+x++xn=1xn+11xs_n=\sum_{i=1}^{n} x^n=1+x+\cdots+x^n=\frac{1-x^{n+1}}{1-x}
  167. Example (Harmonic series) n=11n=1+12+13+=1+(12+13)=24+(14+15+16+17)48+(18+)816\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 cn\sum c_n be convergent series, dn\sum d_n be divergent series. Let an\sum a_n be some series we wish to analyze. If N\exists N s.t. n>N\forall n\gt N, ancn|a_n|\leq c_n, then an\sum a_n converges. If N\exists N s.t. n>N\forall n\gt N, andn|a_n|\geq d_n, then an\sum a_n diverges.
  169. Prop (Tail test) i=1ai\sum_{i=1}^{\infty} a_i converges iff all its tails i=mai\sum_{i=m}^{\infty} a_i converge
  170. Prop (Cauchy condensation test) Suppose ana_n is non-negative and monotone non-increasing. Then n=1an\sum_{n=1}^{\infty} a_n converges iff k=02ka2k\sum_{k=0}^{\infty} 2^k a_{2^k} converges
  171. Prop n=11ns\sum_{n=1}^{\infty} \frac{1}{n^s} converges iff s>1s>1
  172. Prop (Root test) If limsupnann<1\mathop{\lim \sup}_{n\to \infty}\sqrt[n]{|a_n|}\lt 1, then k=1ak\sum_{k=1}^{\infty} a_k converges. If limsupnann>1\mathop{\lim \sup}_{n\to \infty}\sqrt[n]{|a_n|}\gt 1, then k=1ak\sum_{k=1}^{\infty} a_k diverges
  173. Prop (Ratio test) If limsupnan+1an<1\mathop{\lim \sup}_{n\to \infty} \left| \frac{a_{n+1}}{a_{n}} \right|\lt 1, an\sum a_n converges. If liminfnan+1an>1\mathop{\lim \inf}_{n\to \infty} \left| \frac{a_{n+1}}{a_{n}} \right|\gt 1, an\sum a_n diverges
  174. Prop (Alternating series test) Let ana_n be a sequence with alternating signs and suppose an|a_n| are monotone decreasing. Then an\sum a_n converges iff liman=0\lim a_n=0
  175. Thm Suppose a1,a2,a_1, a_2, \dots is a sequence s.t. i=1ai\sum_{i=1}^{\infty} |a_i| converges (absolute convergent), then ai\sum a_i converges, and if aia_i' is any arrangement of aia_i, then ai\sum a_i' converges to ai\sum a_i. If ai\sum a_i is convergent but not absolute convergent, I can find a rearrangement which diverges and a rearrangement which sums to xx xR\forall x\in \mathbb{R}
  176. Thm (Weierstrass M-test) Let M1,M2,M_1, M_2, \dots a sequence of non-negative reals s.t. i=1Mi\sum_{i=1}^{\infty} M_i converges, and f1,f2,f_1, f_2, \dots a sequence of functions XRX\to \mathbb{R} s.t. xX,fi(x)Mi\forall x\in X, |f_i(x)|\leq M_i. Then i=0fi\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 ff is differentiable of xXx\in X if for all sequences t1,t2,xt_1, t_2, \dots \to x, limtxf(t)f(x)tx\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)f'(x)
  179. Rmk If the limit exists, limtxf(t)f(x)=0\lim_{t\to x} f(t)-f(x)=0, so ff differentiable at xx     \implies ff continuous at xx, but not     \impliedby (e.g., corner)
  180. Prop Let f:(a,b)Rf:(a, b)\to \mathbb{R}, let xx be an maximum, i.e., y(a,b),f(x)f(y)\forall y\in (a, b), f(x)\geq f(y), and suppose ff is differentiable at xx. Then f(x)=0f'(x)=0
  181. Lemma If ff is continuous on [a,b][a, b] and f(a)=f(b)f(a)=f(b), then ff has an extremum in (a,b)(a, b)
  182. Thm (Mean Value Theorem) Let f:[a,b]Rf: [a,b]\to \mathbb{R} be continuous, differentiable on (a,b)(a, b). x(a,b)\exists x^{*}\in (a, b) s.t. f(x)=f(b)f(a)baf'(x^{*})=\frac{f(b)-f(a)}{b-a} (Note h(x)=(f(b)f(a))x(ba)f(x)th(x)=\frac{(f(b)-f(a))x-(b-a)f(x)}{t})
  183. Def A partition of [a,b][a, b] is a "finite set of real numbers": a=x0x1xn=ba=x_0\leq x_1\leq \cdots \leq x_n=b, Δxi=xi+1xi\Delta x_i=x_{i+1}-x_i
  184. Def Riemann upper integral of ff on [a,b][a, b] is infPU(P,f)\inf_P U(P, f), lower integral of ff on [a,b][a, b] is supPL(P,F)\sup_P L(P, F)
  185. Prop L(P1,f)U(P2,f)L(P_1, f)\leq U(P_2, f) for every pair of partitions P1,P2P_1, P_2
  186. Def We say PP^* is a refinement of PP if every breakpoint of PP is a breakpoint of PP^*
  187. Lemma If PP^* a refinement of PP, then L(P,f)L(P,f)U(P,f)U(P,f)L(P, f)\leq L(P^*, f)\leq U(P^*, f)\leq U(P, f)
  188. Fact If P1P_1 and P2P_2 are partitions, there is a partition QQ which is a common refinement of both (simply take the union of breakpoints(P1)breakpoints(P2)\text{breakpoints}(P_1)\cup \text{breakpoints}(P_2)), L(P1,f)L(Q,f)U(Q,f)U(P2,f)L(P_1, f)\leq L(Q, f)\leq U(Q, f)\leq U(P_2, f)
  189. Def If lower integral = upper integral (infPU(P,f)=supPL(P,f)\inf_P U(P, f)=\sup_P L(P, f)), we say ff is Riemann integrable on [a,b][a, b] and define abf(x) dx\int_a^b f(x) ~dx to be the common value
  190. Rmk (Dirichlet function) χQ(x)={1xQ0xQ\chi_{\mathbb {Q}}(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}} with χQ:[a,b]{0,1}\chi_{\mathbb{Q}}: [a, b]\to \{0, 1\}, is non-integrable (U(P,χQ)=ba,L(P,χQ)=0U(P, \chi_{\mathbb{Q}})=b-a, L(P, \chi_{\mathbb{Q}})=0)
  191. Rmk χCantor set(x)={1xCantor set0xCantor set\chi_{\text{Cantor set}}(x)={\begin{cases}1&x\in \text{Cantor set} \\0&x\notin \text{Cantor set} \end{cases}} with χCantor set:[0,1]{0,1}\chi_{\text{Cantor set}}: [0, 1]\to \{0, 1\}, is integrable with 01χCantor set(x) dx=0\int_0^1 \chi_{\text{Cantor set}}(x) ~dx=0 (U(Pk,f)=(23)k+ϵ,infPU(P,f)infkU(Pk,f)=0U(P_k, f)=(\frac{2}{3})^k+\epsilon, \inf_P U(P, f)\leq \inf_k U(P_k, f)=0)
  192. Thm If ff is continuous, then ff is integrable
  193. Thm If ff is (bounded) monotone, then ff is integrable
  194. Fact
    1. If c[a,b]c\in [a, b] and ff integrable on [a,b][a, b] then abf(x) dxacf(x) dx=cbf(x) dx\int_a^b f(x) ~dx-\int_a^c f(x) ~dx=\int_c^b f(x) ~dx
    2. If ff is integrable, then f|f| is integrable, and abf(x) dxabf(x) dx\int_a^b |f(x)| ~dx\geq |\int_a^b f(x) ~dx| (triangle inequality)
  195. Def (Indefinite integral) Let ff integrable on [a,b][a, b], then we define F:[a,b]RF: [a, b]\to \mathbb{R} by F(x)=axf(t) dtF(x)=\int_a^x f(t) ~dt
  196. Thm FF is continuous and Lipschitz
  197. Thm (Fundamental Theorem of Calculus, FToC) If ff is continuous at x0[a,b]x_0\in [a, b] and integrable, then FF is differentiable at x0x_0, and F(x0)=f(x0)F'(x_0)=f(x_0)