Introduction to Analysis I: Lecture Notes and Videos

Math 401 (Fall 2021)  - Lecture Notes and Videos on Introduction to Analysis I

 Copyright: I (Bala Krishnamoorthy) hold the copyright for all lecture scribes/notes, documents, and other materials including videos posted on these course web pages. These materials might not be used for commercial purposes without my consent.

Scribes from all lectures so far (as a single big file)

Lec # Date Topic(s) Scribe Panopto
1 Aug 24 syllabus, notation for logical statements, contrapositive proof, proof by contradiction, inductive proof Scb1 Vid1
2 Aug 26 room mixup, sets and operations, union, intersection, distributive laws, set difference, De Morgan's laws Scb2 Vid2
3 Aug 31 Cartesian product, families of sets, set operations over families, functions, composition, image, preimage Scb3 Vid3
4 Sep   2 pre/images commuting with $$\cup,\cap$$, injective, surjective, bijective $$f$$'s, relation, equivalence relation, partition Scb4 Vid4
5 Sep   7 $$\{[x]\}~\forall x \in X$$ partitions $$X$$, equivalence classes of fruits, countability, Cartesian product of countable sets Scb5 Vid5
6 Sep   9 $$\mathbb{Q}$$ is countable, $$\mathbb{R}$$ is uncountable, distances, triangle inequality, convergence, $$\{x_n\} \to a \Rightarrow \{Mx_n\} \to Ma$$ Scb6 Vid6
7 Sep 14 convergence in $$\mathbb{R}^m$$, continuity, $$f_i$$ continuous $$\Rightarrow$$$$f_1+f_2-f_3$$ is, $$g(x)$$ continuous & $$g(a)\neq 0\Rightarrow$$$$1/g(x)$$ is Scb7 Vid7
8 Sep 16 completeness, monotone/bounded sequence, sup, inf, lim sup, lim inf, $$\lim\sup a_n$$$$=$$$$\lim\inf a_n$$$$=$$$$b \Leftrightarrow \lim a_n$$$$=$$$$b$$ Scb8 Vid8
9 Sep 21 Prob 5 in Homework 3, Cauchy sequences converge, continuity using sequences, intermediate value theorem Scb9 Vid9
10 Sep 23 proof of intermediate value theorem, subsequence, Bolzano-Weierstrass theorem, extreme value theorem Scb10 Vid10
11 Sep 28 Rolle's theorem, mean value theorem:$$\,\exists c \in [a,b]$$$$\,:\,$$$$f'(c)$$$$=$$$$(f(b)-f(a)/(b-a)$$, metric spaces, taxicab distance Scb11 Vid11
12 Sep 30 metric has to be finite, more examples of metric spaces, distance between functions, isometry, embedding Scb12 Vid12
13 Oct   5 convergence in metric space, $$\epsilon$$-$$\delta$$ & open ball definitions of continuity of $$f : X \to Y,$$ inverse triangle inequality Scb13 Vid13
14 Oct   7 interior, boundary, and exterior points; open and closed sets, interior and closure of $$A$$, review for midterm Scb14 Vid14
15 Oct 12 Midterm exam
16 Oct 14 $$\bar{A}$$ is closed, $$\overline{B}(\mathbf{a};r)$$ is closed, $$(\bar{A})^{\rm c} = (A^c)^{\rm o}$$, continuity with open sets: $$\forall V \ni f(x_0), \exists U \ni x_0 : f(U) \subseteq V$$ Scb16 Vid16
17 Oct 19 convergent$$\Rightarrow$$Cauchy, complete metric spaces, $$(A,d_A)$$ complete$$\,\Longleftrightarrow A \subset X$$ closed, fixed point, contraction Scb17 Vid17
18 Oct 21 Banach's fixed point theorem, $$f^{\circ n}(x)$$$$=$$$$f(f(\cdots(f(x))\cdots)$$, subsequence in $$(X,d)$$, compact subset of $$(X,d)$$ Scb18 Vid18
19 Oct 26 compact$$\Rightarrow$$closed and bounded, converse in $$\mathbb{R}^m$$, compact$$\Rightarrow$$complete, $$f$$ contns, $$K\,$$compact $$\Rightarrow$$$$f(K)$$ compact Scb19 Vid19
20 Oct 28 extreme value theorem, totally bounded, closed+totally bounded+complete$$\equiv$$compact, open cover property(OCP) Scb20 Vid20
21 Nov  2 OCP$$\implies$$compact, $$f(x)=\sup\{r|B(x,r) \subset O\}$$ continuous, OCP $$\Leftrightarrow$$ compact, problems using OCP Scb21 Vid21
22 Nov  4 pointwise and uniform continuity of functions, equicontinuty, pointwise vs uniform convergence of $$\{f_n\}$$ Scb22 Vid22
23 Nov  9 continuous $$\{f_n(x)\}$$$$\to$$$$f(x)$$ uniformly $$\Rightarrow (f(x)$$ continuous, test for uniform convergence, $$\{\int f_n(x) \}$$$$\to$$$$\int f(x)$$ Scb23 Vid23
Nov 11 No class (Veterans Day)
24 Nov 16 $$\{f_n(x)\}\to$$uniformly$$\implies$$$$\{\int f_n(t)dt\} \to$$uniformly, convergence of series of functions, Weierstrass' M-test Scb24 Vid24
25 Nov 18 differentiating series, space of bounded functions $$B((X,Y),\rho)$$, convergence and completeness in $$B((X,y),\rho)$$ Scb25 Vid25
26 Nov 30 $$C_b(X,Y)$$$$\subseteq$$$$B(X,Y)$$ and closed, $$X$$ compact $$\Rightarrow$$ continuous $$f:X \to Y$$ is bounded, initial value problem (IVP) Scb26 Vid26
27 Dec   2 uniformly Lipschitz, BFPT for IVP, multiple solutions to IVP, dense subset of $$(X,d)$$, separable, $$\mathbb{R}^n$$ is separable Scb27 Vid27
28 Dec   7 compact$$\Rightarrow$$separable, Arzelà-Ascoli theorem: $$K \subseteq C(X,\mathbb{R}^m)$$ compact $$\Leftrightarrow$$ closed, bounded, and equicontinuous Scb28 Vid28
29 Dec   9 $$C([0,1],\mathbb{R})$$ not locally compact, Cauchy-Schwarz inequality (CSI), $$\Delta$$-inequality using CSI, Young's inequality Scb29 Vid29