| Relevance of Course Objectives and Core Learning Outcomes(%) |
Teaching and Assessment Methods for Course Objectives |
| Course Objectives |
Competency Indicators |
Ratio(%) |
Teaching Methods |
Assessment Methods |
| Learn Product measure, Radon-Nikodym theorem, Fourier transforms, and basic properties about Banach spaces and Hilbert spaces. Extend these theories to general knowledge of probability theory. |
| 1.Mathematical Thinking and Logic |
| 2.Professional Knowledge in Mathematical Analysis |
|
|
| topic Discussion/Production |
| Discussion |
| Lecturing |
|
| Written Presentation |
| Attendance |
| Assignment |
| Oral Presentation |
|
| Course Content and Homework/Schedule/Tests Schedule |
| Week |
Course Content |
| Week 1 |
Product space: Product measure, Fubini theorem (Feb. 19) |
| Week 2 |
Product space: Convolution (Feb. 26) |
| Week 3 |
The Radon-Nikodym theorem: Hahn decomposition theorem, The Radon-Nikodym theorem(Mar.5) (Online Class) |
| Week 4 |
The Radon-Nikodym theorem: Lebesgue decomposition theorem, Bounded linear functional(MIDreport1)
(Mar.12) |
| Week 5 |
Fourier transforms: The inversion theorem, The Plancherel theorem
(Mar.19) |
| Week 6 |
Banach spaces: The Hahn-Banach theorem, Baire’s theorem and consequences (MID report2)
(Mar.26) |
| Week 7 |
Holiday(Observed holiday for NCHU Anniversary & Sports meet, no class)(Apr.2) |
| Week 8 |
Hilbert spaces: Inner products, Orthonormal sets (Apr.9) |
| Week 9 |
Hilbert spaces: Frouirer series (Apr.16) |
| Week 10 |
MIDTERM (Apr.23) |
| Week 11 |
Probability: Definition, Independence (Apr.30) |
| Week 12 |
Probability: Weak/Strong law of large numbers(Presentation), Conditional expectation(Presentation)(May 7) |
| Week 13 |
Probability: Martingales(Lecture/Presentation), Weak convergence(Presentation) (May 14) |
| Week 14 |
Probability: Characteristic functions(Lecture/Presentation) (May 21) |
| Week 15 |
Probability: Central Limit theorem&Kolomogorov extenstion(Presentation) (May 28) |
| Week 16 |
Probability: Brownian motion(Lecutre/Presentation) (June 4) |
| Week 17 |
Self-directed learning:
Read related to the probability theory which will be given in class, and submit report.(June 11) |
| Week 18 |
Self-directed learning:
Read related to the probability theory which will be given in class, and submit report.(June 18) |
|
| Evaluation |
Midterm 30% Reports 20%
Final presentation 30%
Self-directed learning HW 20% |
| Textbook & other References |
Real Analysis for Graduate Students, Version 4.2, Richard F. Bass
Real and Complex Analysis(3rd ed.), Rudin, Walter, McGraw-Hill, 1986.
Measures, Integrals and Martingales (2nd ed.), Rene Schilling, Cambridge University Press |
| Teaching Aids & Teacher's Website |
| Lecture note will be distributed in the class. |
| Office Hours |
Wednesday 16:00-17:00
Thursday 16:00-17:00
|
| Sustainable Development Goals, SDGs(Link URL) |
| include experience courses：N |
|