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

Your comments and criticism are greatly welcomed.

# 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 |