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 )
*

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