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

NTU (Spring 2023)

Course Syllabus

Time: Monday, 9:10am-12:10pm, at Social Sciences 609 (社科609)

Office Hour: 9:10-10:00am in class or by Email appointment

TA: Zong-Hong Cheng (鄭宗弘), Danny Po-Hsien Kang (康柏賢), Sean Lan (藍士恩)

Textbook and Recommended Reading:

    1. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw Hill. (Textbook)
    2. Tao, Analysis I: Third Edition, Springer. (e-book available through NTU library)
    3. Protter and Morrey, A First Course in Real Analysis, 2nd ed., Springer.
    4. Interactive Real Analysis. (online book)

Online Resources:

1. Su's Lecture Videos

2. Su's Lecture Notes

3. TA Videos

Class Topics:

 1. [2/20] Lecture 01: Constructing the Rational Numbers (Lecture note 01)
                 Lecture 02: Properties of Q (Lecture note 02)
                     Week 1 Review (video)
                     Quiz 1 and its solution

 o. [2/27] National Holiday

 2. [3/ 6 ] Lecture 03: Construction of R (Lecture note 03)
                 Lecture 04: The Least Upper Bound Property (Lecture note 04)
                     Week 2 Review (video)
                     Quiz 2 and its solution (notes, video)
         

 3. [3/13] Lecture 05: Complex Numbers (Lecture note 05)
                 Lecture 06: The Principle of Induction (Lecture note 06)
                     Week 3 Review (video)
                     Quiz 3 and its solution (notes, video1, video2)
  

 4. [3/20] Lecture 07: Countable/Uncountable Set (Lecture note 07)
                 Lecture 08: Cantor Diagonalization, Metric Space (Lecture note 08)
                     Week 4 Review (video)
                     Quiz 4 and its solution (notes, video)
  

 5. [3/27] Lecture 09: Limit Points (Lecture note 09)
                 Lecture 10: Relationship between Open and Closed Sets (Lecture note 10)
                     Week 5 Review (video)
                     Quiz 5 and its solution (notes, video)

 o. [4/ 3 ] Spring Break

 6. [4/10] Lecture 11: Compact Sets (Lecture note 11)
                 Lecture 12: Relationship between Compact, Closed Sets (Lecture note 12)
                     Week 6 Review (video)
                     Quiz 6 and its solution (notes, video)

 7. [4/17] 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)
                     Week 7 Review (video)
                     Quiz 7 and its solution (notes, video)

 8. [4/24] Lecture 15: Convergence of Sequences (Lecture note 15)
                 Lecture 16: Subsequences, Cauchy Sequences (Lecture note 16)
                     Week 8 Review/Preview (review video; preview video)
                     Quiz 8 and its solution

 9. [5/ 1 ] Lecture 17: Complete Spaces (Lecture note 17)
                 Lecture 18: Series (Lecture note 18)
                     Week 9 Review/Preview (review video; preview video)
                     Quiz 9 and its solution

10.[5/ 8 ] Lecture 19: Series Convergence Tests (Lecture note 19)
                 Lecture 20: Functions - Limits and Continuity (Lecture note 20)
                     Week 10 Review/Preview (review video; preview video)
                     Quiz 10 and its solution

11.[5/15] Lecture 21: Continuous Functions (Lecture note 21)
                 Lecture 22: Uniform Continuity (Lecture note 22)
                     Week 11 Review/Preview (review video; preview video)
                     Quiz 11 and its solution

12.[5/22] Lecture 23: Discontinuous Functions (Lecture note 23)
                 Lecture 24: The Derivative, Mean Value Theorem (Lecture note 24)
                     Week 12 Review/Preview (review video; preview video)
                     Quiz 12 and its solution

13.[5/29] Lecture 25: Taylor's Theorem (Lecture note 25)
                     Week 13 Review (video)
                      Quiz 13 and its solution

14.[6/ 5 ] Final Exam (In-person)
                 Lecture 26: Sequences of Functions (optional) (Su's Lecture note 26)
                 Lecture 27: Brower’s Fixed-Point Theorem (optional) (Su's Lecture note 27)