Notes on Math 521: Analysis I @ University of Wisconsin-Madison
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Course Website
Textbook
Rudin, W. Principles of Mathematical Analysis. Third Edition
Lecture Notes
| Week 1 | |
|---|---|
| 1/24 | Number Systems, Irrationality of √2 |
| 1/26 | Sets, Gaps in Q, Field |
| Week 2 | |
| 1/29 | Field, Order, Upper Bound and Lower Bound |
| 1/31 | Infimum and Supremum, Ordered Field |
| 2/2 | Ordered Field, Archimedean Property, Density of Q in R |
| Week 3 | |
| 2/5 | n-th Root of Real Number, Complex Numbers |
| 2/7 | Complex Numbers, Euclidean Spaces |
| 2/9 | Quiz |
| Week 4 | |
| 2/12 | Schwarz Inequality, Function, Cardinality |
| 2/14 | Finite and Infinite, Sequence |
| 2/16 | Set Operations, Countable and Uncountable |
| Week 5 | |
| 2/19 | Metric Space, Interval, Cell, Ball, Convex |
| 2/21 | Definitions in Metric Space |
| 2/23 | Neighborhood, Open and Closed, De Morgan’s Law |
| Week 6 | |
| 2/26 | Open and Closed, Closure |
| 2/28 | Convergence and Divergence, Range, Bounded |
| 3/2 | Important Properties of Convergent Sequences |
| Week 7 | |
| 3/5 | Algebraic Limit Theorem |
| 3/7 | Convergence of Sequences in R^n, Compact Set |
| 3/9 | Exam 1 |
| Week 8 | |
| 3/12 | Compact Subset, Cantor’s Intersection Theorem |
| 3/14 | Nested Intervals Theorem, Compactness of k-cell |
| 3/16 | Heine-Borel, Weierstrass, Subsequence |
| Week 9 | |
| 3/19 | Cauchy Sequence, Diameter |
| 3/21 | Cauchy Sequence, Complete Metric Space, Monotonic |
| 3/23 | Upper and Lower Limits |
| Week 10 | |
| 4/2 | Some Special Sequences |
| 4/4 | Series, Cauchy Criterion for Series, Comparison Test |
| 4/6 | Convergence Tests for Series |
| Week 11 | |
| 4/9 | Power Series, Absolute Convergence, Rearrangement |
| 4/11 | Rearrangement, Limit of Functions |
| 4/13 | Exam 2 |
| Week 12 | |
| 4/16 | Continuous Function and Open Set |
| 4/18 | Continuity and Compactness, Extreme Value Theorem |
| 4/20 | Uniform Continuity and Compactness |
| Week 13 | |
| 4/23 | Connected Set, Intermediate Value Theorem |
| 4/25 | Derivative, Chain Rule, Local Extrema |
| 4/27 | Mean Value Theorem, Monotonicity, Taylor’s Theorem |
| Week 14 | |
| 4/30 | Riemann-Stieltjes Integral, Refinement |
| 5/2 | Fundamental Theorem of Calculus |
| 5/4 | Sequence of Functions, Uniform Convergence |
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