Introduction to Real Analysis (Online Math Camp 分析導論)

NTU (Fall 2024)

Course Syllabus

TA: Danny Po-Hsien Kang (康柏賢) (Office hour by email appointment)

Textbook and Recommended Reading:

    1. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw Hill. (Textbook)
    2. Ok, Real Analysis with Economic Applications, Princeton University Press. (Chapter A)
    3. Tao, Analysis I: Third Edition, Springer. (e-book available through NTU library)
    4. Interactive Real Analysis. (online book)

Online Resources:

1. Su's Lecture Videos

2. Su's Lecture Notes

3. TA Videos

Class Topics:

 1. Lecture 01: Constructing the Rational Numbers (Lecture note 01)
     Lecture 02: Properties of Q (Lecture note 02)
     Lecture 1-2 Review (video)
     Exercise: Self Quiz 1 and its solution

 2. Lecture 03: Construction of R (Lecture note 03)
     Lecture 04: The Least Upper Bound Property (Lecture note 04)
     Lecture 3-4 Review (video)
     Exercise: Self Quiz 2 and its solution (notes, video)
         

 3. Lecture 05: Complex Numbers (Lecture note 05)
     Lecture 06: The Principle of Induction (Lecture note 06)
     Lecture 5-6 Review (video)
     Exercise: Self Quiz 3 and its solution (notes, video1, video2)
  

 4. Lecture 07: Countable/Uncountable Set (Lecture note 07)
     Lecture 08: Cantor Diagonalization, Metric Space (Lecture note 08)
     Lecture 7-8 Review (video)
     Exercise: Self Quiz 4 and its solution (notes, video)
  

 5. Lecture 09: Limit Points (Lecture note 09)
     Lecture 10: Relationship between Open and Closed Sets (Lecture note 10)
     Lecture 9-10 Review (video)
     Exercise: Self-Quiz 5 and its solution (notes, video)

 6. Lecture 11: Compact Sets (Lecture note 11)
     Lecture 12: Relationship between Compact, Closed Sets (Lecture note 12)
     Lecture 11-12 Review (video)
     Exercise: Self Quiz 6 and its solution (notes, video)

 7. Lecture 13: Compactness, Heine-Borel Theorem (Lecture note 13/Su's Lecture note 12, 13)
     Lecture 14: Connected Sets, Cantor Sets (Lecture note 14/Su's Lecture note 13, 14)
     Lecture 13-14 Review (video)
     Exercise: Self Quiz 7 and its solution (notes, video)

 8. Lecture 15: Convergence of Sequences (Lecture note 15)
     Lecture 16: Subsequences, Cauchy Sequences (Lecture note 16)
     Lecture 15-16 Review/Lecture 17-18 Preview (review video; preview video)
     Exercise: Self Quiz 8 and its solution

 9. Lecture 17: Complete Spaces (Lecture note 17)
     Lecture 18: Series (Lecture note 18)
     Lecture 17-18 Review/Lecture 19-20 Preview (review video; preview video)
     Exercise: Self Quiz 9 and its solution

10.Lecture 19: Series Convergence Tests (Lecture note 19)
     Lecture 20: Functions - Limits and Continuity (Lecture note 20)
     Lecture 19-20 Review/Lecture 21-22 Preview (review video; preview video)
     Exercise: Self Quiz 10 and its solution

11.Lecture 21: Continuous Functions (Lecture note 21)
     Lecture 22: Uniform Continuity (Lecture note 22)
     Lecture 21-22 Review/Lecture 23-24 Preview (review video; preview video)
     Exercise: Self Quiz 11 and its solution

12.Lecture 23: Discontinuous Functions (Lecture note 23)
     Lecture 24: The Derivative, Mean Value Theorem (Lecture note 24)
     Lecture 23-24 Review/Lecture 25-26 Preview (review video; preview video)
     Exercise: Self Quiz 12 and its solution

13.Lecture 25: Taylor's Theorem (Lecture note 25)
     Lecture 25 Review (video)
     Exercise: Self Quiz 13 and its solution

14.Lecture 26: Sequences of Functions (optional) (Su's Lecture note 26)
     Lecture 27: Brower’s Fixed-Point Theorem (optional) (Su's Lecture note 27)

15.Final Exam (In-person)