Spring 2018, Math 811, Graduate Real Analysis II
This course is a continuation of graduate real analysis, math 810. In math 810, we should have covered measures and
integration, signed measures and the Radon-Nikodym theorem, L^p spaces. Basically it is the content of Chapter 1,2,3,6 in
the book by G. Folland, ``Real Analysis: Modern techniques and their applications". In this course, we tentatively plan to cover
three parts. In the first part, we tentatively plan to cover Folland's book Chapter 4 (point set topology), Chapter 5
(Elements of functional analysis), and Chapter 7 (Radon measures). This is mainly concerned with Urysohn's lemma, Tietze's extension
theorem,LCH spaces, Tychonoff's theorem, Arzela-Ascoli's theorem, the Stone-Weierstrass theorem, the Hahn-Banach theorem,
Baire Category theorem, the open mapping theorem, the uniform boundedness principle, and Alaoglu's theorem, and the Riesz's representation
theorem for positive linear functionals and on C_0.
In the second part, we will also
introduce Chapter 8 (Fourier Analysis) and Chapter 9 (Distribution theory). In this part, we supplement the first
chapter of the book by E. Stein and G. Weiss, ``Introduction to Fourier Analysis on Euclidean
spaces". This is mainly concerned with the L^1 and L^2 theory of the Fourier transform, and the distribution.
In the third part, we will introduce the extremal problem for Fourier restriction inequalities in harmonic analysis
such as the Strichartz inequalities for the Schrodinger equations.
Time: MWF 2:00 --- 2:50 PM
Location: Snow Hall 456
- Instructor: Shuanglin Shao
- Office: Snow Hall 615
- Email: email@example.com
- Office Hour: MWF: 1:00 --- 2:00 PM, or by appointment.
- Homework 1.(due by 02/12 in class.): Chapter 4. 1,3,11, 14,16,21,23,30,31,36, 39,40,43.
- Homework 2.(due by 02/26 in class.): Chapter 4. 54,56,64,65,68,70.
- Homework 3.(due by 03/12 in class.): Chapter 5. 7,11,17,19,29,30,45,48,57,58.
- Homework 4.(due by 03/26 in class.): Chapter 7. 1,2,7,8,17.
- Homework 5.(due by 04/09 in class.): Chapter 8. 1,6,7,8,14,18,26,27,30.
- Homework 6.(due by 04/23 in class.): Chapter 9.10,11,14(a)(b)(c)(d),27.