MATH 245B : Real Analysis


         (Jan 14) Note that there are errata for some Folland questions, in some printings of Folland; see this page.(For instance, Q17 of Chapter 3 has a misprint in the first five printings.)

         (Jan 25) It has come to my attention that some of the solutions submitted for HW questions have been copied more or less directly from the internet or similar sources.While I encourage attempts to work together or research these problems by a variety of means, I would like at least to see the solutions expressed in oneís own words, to demonstrate innate understanding of the solution.Accordingly, I am asking Tim, whenever he sees two or more solutions that are essentially identical in terms of notation, logical structure, wording, and minor errors, to split the points for that question equally among all students submitting that solution.[It should also be noted that attempting the homework questions unassisted will be excellent practice for the midterm and final; conversely, if you find yourself unable to do a majority of the HW questions without assistance, this does not bode particularly well for your performance at these exams.]

         (Feb 4)Midterm solutions are available here.

         (Mar 2) The final has been changed from a 3-hour examination to a take home format, to be distributed on Friday and returned on Monday, as I will be out of the country on the scheduled day of the final.More details below.You can use any books, notes, or on-line references to assist you, but if you use any assistance outside of the textbook, lecture notes, or your own notes, then they must be cited, or the points for any solution using the citation will be forfeited.As with the homework, if two or more solutions to a given problem are essentially identical, then the points for these problems will be shared equally between the solutions.

         Instructor: Terence Tao,, x64844, MS 6183

         Lecture: MWF 10-10:50, MS 5117.††

         Discussion section: Th 10-10:50, MS 5117

         Office Hours: Tu 2-2:50, W 1-2

         TA: Tim Austin, MS 3931.See also Timís web page for this course.

         TA Office hours: Tu 11:30-12:30; Thu 1:30-2:30

         Textbook: Folland, Real Analysis, Second Edition; we will also use Stein-Shakarchiís Real Analysis as a supplementary text.We will cover the Radon-Nikodym theorem from Chapter 3 of Folland (or Chapter 6.4 of Stein-Shakarachi), as well as large parts of Chapters 4-6 of Folland (Point Set Topology, Functional Analysis, L^p spaces); some variation from this plan may develop depending on time constraints.I will also write some supplemental lecture notes to these texts on my blog.

         Prerequisite: Math 245A.The material here will rely heavily on that in Math 245A, so be sure you have a firm grasp of the material from that course.

         Grading: Homework (30%), Midterm (30%), Final (40%).In addition, each student must present at least one homework problem at the blackboard during at least one discussion section in order to gain the Homework portion of the grade.†† You may use any resource to solve homework problems, so long as the final solution is in your own words; duplicate solutions will have the points for that question shared equally between them.For some of the homeworks, due to time constraints, only a subset of the questions (chosen at random) will be graded in full.

         Exams: The midterm will be on Wednesday, Feb 4, in MS 5117, and will consist of three questions of a level comparable to the HW, covering measure theory, L^p spaces, Hilbert spaces, the Hahn-Banach theorem, and the basic theory of normed vector spaces.The optional notes on the blog will not be covered.Some sample questions (taken from my previous course web pages) are given at this page.The exam is open book and open notes (and any result from lecture, notes, or text can be cited freely), but no electronic devices will be allowed.The final will be a take-home exam of approximately seven questions (comparable to that of the homework, and focusing on the material covered in second half of the quarter), handed out in the final class on Friday March 13, and due before noon, Monday March 16, to my office or (less preferably) in my mailbox.Sample exams can be found at my web pages for previous 245B classes.

         Homework: There will be approximately seven homework assignments, due in discussion sections.There will be roughly 10 questions in each assignment; unless otherwise indicated, they will be from Folland.

         Assignment 1 (Due Thu, Jan 15): Folland Chapter 3, Q 3, 7, 10, 11, 17; Chapter 5, Q 1, 5, 9

         Assignment 2 (Due Thu, Jan 22): Folland Chapter 6, Q 10, 15; Chapter 5, Q 7, 11

         Assignment 3 (Due Thu, Jan 29): Folland Chapter 5, Q 19, 20, 56, 62, 67; Chapter 6, Q 6, 12

         Assignment 4 (Due Thu, Feb 12): Folland Chapter 5, Q 22c, 30, 32, 42

         Assignment 5 (Due Fri, Feb 20): Folland Chapter 4, Q28ab, 43, 59, 65

         Assignment 6 (Due Thu, Feb 26): Folland Chapter 4, Q63; Chapter 5, Q47, 48, 51

         Assignment 7 (Due Thu, Mar 5): Folland, Chapter 4, Q3, 46, 68; Chapter 5, Q53