Math 521

Notes on Math 521: Analysis I @ University of Wisconsin-Madison

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Rudin, W. Principles of Mathematical Analysis. Third Edition

Lecture Notes

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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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