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