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

Last Modified: Thu Dec 09 19:52:37 PST 2021