National Taiwan Normal University Course Outline
Spring , 2019

@尊重智慧財產權,請同學勿隨意影印教科書 。
Please respect the intellectual property rights, and shall not copy the textbooks arbitrarily.

I.Course information
Serial No. 2708
Course Code MAC0088 Chinese Course Name 實變分析(二)
Course Name Real Analysis (II)
Department Department of Mathematics
Two/one semester 1 Req. / Sel. Sel.
Credits 3.0 Lecturing hours Lecture hours: 3
Restrict Course
Comment 授課教師Ulrich Menne
Course Description
Time / Location Mon. 2-4 Gongguan 11111

II. General Syllabus
Instructor(s) Ulrich Menne/ 孟悟理

This course is a continuation of the course Real Analysis I which treated – apart of the necessary preparations from topology, metric spaces, and functional analysis – the following topics: measures, measurable sets, measurable functions, and Lebesgue integration (basic properties and limit theorems). The lecture notes on this course (see [Men18]) will be available and continued.

The core of the course Real Analysis II shall consist of the following topics treated in accordance with [Fed69]:

  1. Lebesgue integration: integrals over subsets, Lebesgue spaces, composition and image measures, Jensen’s inequality.
  2. Linear functionals: lattices of functions, Daniell integrals, linear functionals on Lebesgue spaces, Riesz’s representation theorem.
  3. Product measures: Fubini’s theorem, Lebesgue measure.

Apart of this core, the necessary preparations from topology (locally compact Hausdorff spaces and Tychonoff’s theorem) and functional analysis (Hahn-Banach theorem and Banach-Alaoglu theorem) will be provided following [Kel75] and [DS58], respectively. If time permits, Riesz’s representation theorem shall also be put in the context of the theory of locally convex spaces following [Bou87].
PDF file of this course description including references

Lecturing Methodologies
Methods Notes
Formal lecture Lecture in English with weekly updated lecture notes (LaTeX) in English.
Problem base learning Weekly exercises in English; help session in Mandarin.
Grading assessment
Methods Percentage Notes
Final exam 100 % Individual oral exam conducted in English determining pass or fail as well as the grade for the course. To be eligible to take the oral exam, a student needs to have obtained at least 50% of the maximal score in the weekly exercises to be submitted in English.
Required and Recommended Texts/Readings with References

[Bou87] N. Bourbaki. Topological vector spaces. Chapters 1–5. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1987. Translated from the French by H. G. Eggleston and S. Madan. URL:

[DS58] Nelson Dunford and Jacob T. Schwartz. Linear Operators. I. General Theory. With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. URL:;view=1up;seq=9A.

[Fed69] Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. URL:

[Kel75] John L. Kelley. General topology. Springer-Verlag, New York, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27.

[Men18] Ulrich Menne. Real analysis, 2018. Lecture notes, National Taiwan Normal University.