| 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.Basic Knowledge in Mathematical Sciences |
| 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. 25)
|
| Week 2 |
Product space: Convolution (Mar.4) |
| Week 3 |
The Radon-Nikodym theorem: Hahn decomposition theorem, The Radon-Nikodym theorem (Mar.11)
|
| Week 4 |
The Radon-Nikodym theorem: Lebesgue decomposition theorem, Bounded linear functional(MID report1) (Mar.18) |
| Week 5 |
Fourier transforms: The inversion theorem, The Plancherel theorem (Mar.25)
|
| Week 6 |
Banach spaces: The Hahn-Banach theorem, Baire’s theorem and consequences (MID report2) (Apr.1) |
| Week 7 |
Holiday(Observed holiday for NCHU Anniversary & Sports meet, no class)(Apr.8) |
| Week 8 |
Hilbert spaces: Inner products, Orthonormal sets (Apr.15) |
| Week 9 |
Hilbert spaces: Frouirer series (Apr.22) |
| Week 10 |
MIDTERM (Apr.29) |
| Week 11 |
Probability: Definition, Independence (May 6) |
| Week 12 |
Probability: Weak/Strong law of large numbers(Presentation), Conditional expectation(Presentation) (May 13) |
| Week 13 |
Probability: Martingales(Lecture/Presentation), Weak convergence(Presentation) (May 20) |
| Week 14 |
Probability: Characteristic functions(Lecture/Presentation) (May 27) |
| Week 15 |
Probability: Central Limit theorem&Kolomogorov extenstion(Presentation) (June 3) |
| Week 16 |
Probability: Brownian motion(Lecutre/Presentation) (June 10)
|
self-directed
learning |
05.Participation in various workshops organized by different departments of NCHU.
06.Participation in field trips and outdoor instructional activities at other NCHU campuses or branches, including experimental forests or test sites.
|
|
| Evaluation |
Midterm 40% Reports 20%
Final presentation 40%(20% presentation/20% report of other presentation)
|
| Textbook & other References |
Real Analysis for Graduate Students, Version 4.3, 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 |
|