Real Analysis Key Concepts
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1 Real Analysis Key Concepts Xiangzhu Long Carnegie Mellon University According to Principles of Mathematical Analysis by Walter Rudin (Chapter 1-5) December 8, 2015 Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
2 Overview 1 The Real and Complex Number 2 Basic Topology 3 Numerical Sequences and Series 4 Continuity 5 Differentiation Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
3 Sets Set Operations Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
4 Fields Addition Axioms Mutiplication Axioms Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
5 Boundaries Upper Bound Least Upper Bound Lower Bound Greatest Lower Bound Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
6 Dedekind Cut 1. non-trival 2. closed downwards 3. no largest number eg: α = Q Yes A = {x x 2 < 2, x Q} No Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
7 Numbers Real Numbers (R) Relational Numbers (Q) Natural Numbers (N) Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
8 Relationship bt. R and Q Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
9 Complex Numbers C Operations Conjugate Inner Product Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
10 Induction 1. Base case 2. Inductive steps Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
11 Sets Relationship Injection Surjection Bijection Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
12 Finite Set J n Finite sets: Infinite sets: N Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
13 Countable Set Countable sets: N, Z, Q Uncountable sets: R Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
14 Countability Theorems Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
15 Cantor Theorem Powerset Cantor Theorem: A 2 A Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
16 Metric Spaces 1. non-negativity 2. symmetry 3. triangle inequality eg: Euclidean spaces Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
17 Limit Point Open ball Closed ball Limit point Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
18 Points and Sets Isolated points Interior points Open set Closed set Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
19 Closure Closure of a set Closure property: Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
20 Relationship bt. Open and Closed Sets Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
21 Compact Sets Open Cover Subcove Compact Set Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
22 Bounded Sets Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
23 Compactness Theorems I Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
24 Compactness Theorems II In R n, K is compact K is closed and bounded. Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
25 Finite Interesction Property Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
26 Cantor Set Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
27 Connectness Separated sets Connected sets eg: [a,b] is connected Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
28 Converge Converge Diverge Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
29 Converge Theorems Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
30 Subsequence Subsequence Subsequence theorems Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
31 Cauchy Sequence Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
32 Monotonic Sequence Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
33 Series Series Series Converge Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
34 Series Theorems Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
35 Function Limits lim f (x) = q (ɛ δ ball) x p Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
36 Limit Properties 1. unique 2. lim x p f (n + m) = lim x p f (n) + lim x p f (m) 3. algebraic limit theorem: Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
37 Continuous Function f continous at p lim x p f (x) = f (q) Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
38 Continuous Function on Compact Set I f : X Y is continuous open set U Y, f 1 (U) is open in X, closed set K Y, f 1 (K) is closed in X, Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
39 Continuous Function on Compact Set II f : X Y is continuous, X compact f (x) is compact. Extrem Value Theorem Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
40 Uniform Continuity Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
41 Intermediate Value Theorm If f : [a, b] R continuous, and f (a) < c < f (b), then x (a, b) such that f (x) = c. Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
42 Discontinuous Dirichlet funtion Simple Discontinuity Second Discontinuity Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
43 Monotonic Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
44 Differentiation Xiangzhu Long (CMU) Real Analysis Key Concepts December 8, / 44
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