Real Analysis

This course, Real Analysis, delves into advanced mathematical concepts, building upon foundational knowledge of calculus. It is designed for students seeking a deeper understanding of mathematical theory and its applications. The curriculum covers key areas such as differentiation and integration of real variables, functions of bounded variation, and absolutely continuous functions. It also explores advanced topics like product measures, the change of variables formula in Euclidean space, and applications to potential integrals. Further topics include linear analysis, Hilbert spaces, L^p spaces, Fourier integrals, and Sobolev spaces, with a focus on their applications to the regularity of weak solutions of partial differential equations. Elements of probability theory are also integrated into the course.

Curriculum

The course is structured into 50 lectures covering a wide range of topics in Real Analysis. The lectures are organized into modules focusing on specific areas of study.

1. Part 1
   • IntroductionIntroduction
   • AdditivityAdditivity
   • Measure 1Measure 1
   • Measure 2Measure 2
   • Measure and Integral 1Measure and Integral 1
   • Measure and Integral 2Measure and Integral 2
   • Measure and Integral 3Measure and Integral 3
   • Metric Space 1Metric Space 1
   • Metric Space 2Metric Space 2
   • Outer Measure 1Outer Measure 1
   • Outer Measure 2Outer Measure 2
   • Outer Measure 3Outer Measure 3
   • Outer Measure 4Outer Measure 4
   • Outer Measure 5Outer Measure 5
   • Review Caratheodory MeasureReview Caratheodory Measure
   • Caratheodory Measure 2Caratheodory Measure 2
   • Caratheodory Measure 3Caratheodory Measure 3
   • Caratheodory Measure 4Caratheodory Measure 4
   • Caratheodory Measure 5Caratheodory Measure 5
   • Riemann and Lebesgue IntegralRiemann and Lebesgue Integral
   • Riemann and Lebesgue Integral (2)Riemann and Lebesgue Integral (2)
   • Differentiation of Radon MeasureDifferentiation of Radon Measure
   • Compactness and Measure (1)Compactness and Measure (1)
   • Compactness and Measure (2)Compactness and Measure (2)
   • Compactness and Measure (3)Compactness and Measure (3)
   • Push-forward of measures (1)Push-forward of measures (1)
   • Push-forward of measures (2)Push-forward of measures (2)
   • Lebesgue Decomposition (1)Lebesgue Decomposition (1)
   • Lebesgue Decomposition (2)Lebesgue Decomposition (2)
   • Lebesgue Decomposition (3)Lebesgue Decomposition (3)
   • Product measures and Fubini theoremProduct measures and Fubini theorem
   • Product measures and Fubini theorem (2)Product measures and Fubini theorem (2)
   • Change of variable in Lebesgue integralChange of variable in Lebesgue integral
   • Change of variable in Lebesgue integral (2)Change of variable in Lebesgue integral (2)
   • Change of variable in Lebesgue integral(2-2)Change of variable in Lebesgue integral(2-2)
   • Change of variable in Lebesgue integral (3)Change of variable in Lebesgue integral (3)
   • Polar coordinatesPolar coordinates
   • Change of variable in Lebesgue integral (4)Change of variable in Lebesgue integral (4)
   • Change of variable in Lebesgue integral (5)Change of variable in Lebesgue integral (5)
   • Hilbert SpaceHilbert Space
   • Hilbert Space (2)Hilbert Space (2)
   • Hilbert Space (3)Hilbert Space (3)
   • Hilbert Space (4)Hilbert Space (4)
2. Part 2
   • The Space L2The Space L2
   • The Space L2 (2)The Space L2 (2)
   • The Space L2 (3)The Space L2 (3)
   • The Space L2 (4)The Space L2 (4)
   • The Space L2 (5)The Space L2 (5)
   • The Space L2 (6)The Space L2 (6)
   • The Space L2 (7)The Space L2 (7)

Frequently asked questions

What are the academic entry requirements for the Real Analysis Master's program?

A Bachelor's Degree from a regionally accredited college or university is required for Master's programs. English proficiency requires a CEFR score of B2 (Vantage) or above, unless you are from an English-speaking country or your previous degree was taught in English.

What is the estimated cost of living for international students per semester?

The estimated cost of living per semester, excluding tuition fees, ranges from TWD 84,000 to TWD 113,000 (approximately USD 2,800 – 3,767). This estimate covers books, housing, insurance, and general living expenses.

What are the application deadlines for international students?

For the 2026 February entry, the online application period is from August 4, 2025, to September 18, 2025. For the 2026 September entry, there are two rounds: the first is from October 1, 2025, to November 13, 2025, and the second is from December 1, 2025, to January 15, 2026.

What documents are required for the application?

Required documents include your highest degree's graduation certificate and transcript (with translations if necessary), a study plan, proof of English proficiency, recommendation letters, and a financial statement showing at least NT$200,000 or US$6,000 in savings.

Are there any scholarships available for international students?

The university offers the Taiwan Government Scholarships and the NTU Scholarship for international students. Specific amounts and how to apply are available on the Overseas Student Admissions website and the ISSS Finances > Scholarships page, respectively.