This part of 240 series is about (i) Basic functional analysis; (ii) Properties of L^p spaces; (iii) Locally compact Hausdorff spaces and Radon measures on them; (iv) Basics of distribution theory and fourier analysis which are needed for the study of PDEs and other subjects of mathematics, (v) Sobolev functions, BV functions and convex functions. The first three items form the main topics of Math 240B. The topics in (iv) and (v) are covered in Math 240C. These are very basic topics of analysis which are needed for the further studies of almost all areas of more advanced mathematics. Part (v) shall NOT be covered by the qualifying exams.
Since some basics of the set topology are needed in Math 240B, all students enrolled are required (via the announcement via canvas) to do some readings on set topology covered in Folland's text Ch 4.1, 4.2 and complete at least half of the exercises there before the start of lectures. The alternative is to take math 190A.
Almost all problems in this course require the proof. Any mathematical proof in serious mathematical textbooks as put by A. Y. Khinchin ` will undoubtedly seem very complicated to you. But it will take you only two to three week's work with pencil and paper to understand and digest it completely. It is by conquering difficulties of just this sort, that the mathematicians/or mathematical students grow and develop.' Even though none of the theorems involved in this course is a difficult one requiring the labor beyond several hours to digest, the same principle on the effort part applies. No good mathematics can be spoonfed fast. Learning mathematics also takes time and serious efforts via many practices of solving exercises and homework problems.


Required Text: Real Analysis, by Folland, 2nd ed., Wiley, 1999.
Recommend Texts for possible alternate/different approach on the topics covered: (1) Functional Analysis, by P. Lax, Wiley, 2002;
(2) Analysis, by Lieb and Loss, AMS, 2001;
(3) Real Analysis, 3rd edition, by Royden, Pearson Education, 1988;
(4) Real and Complex Analysis, 3rd edition, by Rudin, McGrawHill, 1987;
(5) Measure theory and fine properties of functions, by Evans and Gariepy, CRC press 1992.
Further readings: (1) Functional Analysis by F. Riesz and B. Nagy, New York, Ungar, 1956;
(2) Lectures on Geometric Measure Theory, by Leon Simon, Centre for Mathematical Analysis, Australian National University, 1984.
Important Resources
Errata for Folland's text: [html]
Errata for Evans/Gariepy's text: [html]
Problem Solving Techniques in Analysis by Terence Tao: [html]
LaTeX homework template: [tex]
Everything you need to know to typeset your homework in LaTeX: [pdf]
(i) Handout1 on L^p spaces; (ii) Handout of lecture notes; (iii) Handout2 on L^p spaces;
(iv) Handout on Sobolev spaces; (v) Handout1 on BV functions; (vi) Handout1 on Fourier Transforms.
(vii) Handout1 on Distributions; (viii) Handout on general Riesz representation theorem; (ix) Handout on BVfunctuions II
Solution to final in Fall 2014
Solution to midterm in Winter 2015
Solution to the midterm of 240C
Content covered by the Qualifying Exam
Solutions to the HWs of 240C (including the all previous solutions)
