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National Taiwan Normal University Course Outline Fall , 2026 |
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| I.Course information |
| Serial No. | 2767 | Course Level | Undergraduate / Master |
| Course Code | MAC0087 | Chinese Course Name | 實變分析（一） |
| Course Name | Real Analysis (I) | ||
| Department | Department of Mathematics | ||
| Two/one semester | 1 | Req. / Sel. | Sel. |
| Credits | 3.0 | Lecturing hours | Lecture hours: 3 |
| Teach in English | Y | Teach in National Languages | |
| Prerequisite Course | |||
| Comment | |||
| Course Description | |||
| Day & Class Period/Location | Mon. 8 Gongguan M310, Thur. 7-8 Gongguan M310 | ||
| Curriculum Goals | Corresponding to the Departmental Core Goal | ||
| 1. Cultivate Mathematics Professional Ability |
College: 1-1 Equipped with professional mathematics competences 2-1 Being able to communicate and express mathematically 3-1 Being able to seek out answers with the attitudes of patience, diligence, concentration, and curiosity 4-2 Possessing a consistent and firm attitude toward pursuing truths Master: 1-1 Equipped with professional mathematics competences 2-1 Being able to communicate and express mathematically 3-1 Being able to seek out answers with the attitudes of patience, diligence, concentration, and curiosity 4-2 Possessing a consistent and firm attitude toward pursuing truths |
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| 2. Pathway to advanced analytics courses |
College: 1-2 Being able to reason and induct with mathematical logic 1-4 Possessing the abilities to propose and solve questions in advanced mathematics 3-2 Possessing the abilities to think independently, criticize, and reflect 4-1 Being knowledgeable and being able to self-develop in the profession Master: 1-2 Being able to reason and induct with mathematical logic 1-4 Possessing the abilities to propose and solve questions in advanced mathematics 3-2 Possessing the abilities to think independently, criticize, and reflect 4-1 Being knowledgeable and being able to self-develop in the profession |
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| 3. Raise the level of abstract thinking |
College: 1-3 Being able to think mathematically and critically 3-4 Having insights, intuitions, and senses of mathematics 4-3 Possessing a variety of beliefs regarding mathematics values and mathematics learning Master: 1-3 Being able to think mathematically and critically 3-4 Having insights, intuitions, and senses of mathematics 4-3 Possessing a variety of beliefs regarding mathematics values and mathematics learning |
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| 4. Interpret the connection between mathematics and other disciplines from a high perspective |
College: 1-5 Being able to use mathematics as tools to learn other subjects 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint 2-4 Possessing the competences of lifelong learning 3-5 Having good taste for mathematics 4-4 Possessing global views from both scientific and humanistic perspectives, and being able to appreciate the values of other knowledge fields Master: 1-5 Being able to use mathematics as tools to learn other subjects 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint 2-4 Possessing the competences of lifelong learning 3-5 Having good taste for mathematics 4-4 Possessing global views from both scientific and humanistic perspectives, and being able to appreciate the values of other knowledge fields |
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| II. General Syllabus |
| Instructor(s) | Riju Basak/ Riju Basak | ||
| Schedule | |||
Lebesgue Measure and Outer Measure: Introduction, Lebesgue outer measure, Lebesgue measurable sets and Lebesgue measure, A nonmeasurable set, Measurable functions, Littlewood’s Three Principles The Lebesgue Integral: The Riemann Integral, The Lebesgue integral of a bounded function over a set of finite measure, The integral of a non-negative function, The general Lebesgue integral, Convergence in measure Differentiation and Integration: Differentiation of Monotone Functions, Functions of Bounded Variation, Differentiation of an integral, Absolute Continuity, Convex Functions The Classical Banach Spaces: The Lp spaces, The Minkowski and Holder inequalities, Convergence and Completeness, Approximation in Lp, Bounded Linear Functionals on the Lp Spaces Measure and Integration: Measure Spaces, Measurable functions, Integration, General Convergence Theorems, Signed Measures, Radon-Nikodym Theorem, The Lp Spaces Measure and Outer Measure: Outer measure and measurability, The Extension theorem, Product measures, Integral operators, Hausdorff measure
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| Instructional Approach | |||
| Methods | Notes | ||
| Formal lecture | The course will be primarily lecture-based, supplemented by exercises and classroom discussions. | ||
| Group discussion | Each week includes a practice and discussion session. | ||
| Grading assessment | |||
| Methods | Percentage | Notes | |
| Assignments | 30 % | Weekly homework assignments | |
| Midterm Exam | 30 % | One Midterm exam | |
| Final exam | 40 % | One final exam | |
| Adjustment methods for students | |||
| Required and Recommended Texts/Readings with References | Material: Primary reference: H.L. Royden, Real Analysis (Third Edition), Chapters 3-6, 11, 12.
Other references: 1. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions. 2. R. L. Wheeden and A. Zygmund, Measure and Integral
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