Textbook abbreviations FD = Folland's Real Analysis LAX =Lax's Functional Analysis RC = Rudin's Real and Complex Analysis EG = Evans and Gariepy's Measure theory and fine properties of functions LL = Lieb and Loss's Analysis

Date Topics Sections Event Week 1 01-10 Normed vector spaces and linear functionals--Hahn-Banach Theorem -real case FD 5.2 and Week1 Notes-Part A Read FD Ch 5.1, LAX Ch 1-2 01-12 Hahn-Banach Theorem -complex case and further applications FD 5.2, LAX 3.2 and Week1 Notes--Part B Read FD 4.1 and Proposition 4.13; LAX 3.2 Week 2 01-17 L^p spaces as Banach spaces, monotonicity and convexity of norms FD 6.1, RC Ch3 and Week2 Notes-Part A Read about $L^\infty$, LAX 5.1, LL Ch 2, FD Ch 3 Exercise 42 01-19 L^p spaces--Duality via complex measures FD 6.2, RC Ch6 and Week2 Notes-Part B Read RC Theorem 6.2, 6.4; FD 3.3; LAX Ch 8.1, 8.2 Week 3 01-24 Lebesgue differentiation theorem FD 3.4 and 6.3 Week3 Notes-Part A Read RC Ch 7 01-26 L^p spaces--Convexity of the norms and the norms of the linear mappings FD 6.4-6.5 Week3 Notes-Part B1 and Week3 Notes-Part B2 Read FD, Ch 6.4, 6.5 Week 4 01-31 The Baire Category Theorem and open mapping theorem FD 5.3, RC Ch5, Week4 Notes-Part A LL 2.11, 2.12 02-02 Applications and weak/weak-star topology FD 5.3, 5.5, Week4 Notes-Part B Read LAX 6.1-6.2, RC Ch 5.2 Week 5 02-07 Product topology and weak topology FD 4.2, 5.4, LAX Ch10, Week5 Notes-Part A Read LAX Ch10 02-09 Midterm exam Midterm exam This week's homework due time is extended for 48 hours Week 6 02-14 Topological vector spaces-Alaoglu's theorem FD 5.4, 4.6, 4.3, Week6 Notes-Part A FD, Proposition 4.21--4.26 02-16 Nets (generalizing sequences) and compactness FD 4.3,4.4, Week6 Notes-Part B Week6 Notes-Part B2   Week 7 02-21 Locally compact Hausdorff spaces FD 4.4, 4.5, Week7 Notes-Part A   02-23 Catch-up/Make-up-A Lebesgue theorem and BV functions of one variable Catch-up/Make-up FD Theorem 3.22, pages 105--107 Week 8 02-27 Hilbert spaces-I FD 5.5, LAX Ch 6, RC Ch4, Week8 Notes-Part A   03-02 Hilbert spaces-II FD 5.5, LAX Ch6, RC Ch4, Week8 Notes-Part B FD 7.1, RC, 2.14 Week 9 03-07 Radon measures-Riesz Representation for positive functionals FD 7.1, Week9 Notes-Part A RC 2.14 03-09 Regularity of the Borel measures FD 7.2, Week9 Notes-Part B RC 2.15-2.18 Week 10 03-14 Catch-ups and Make-ups-Borel regular measures on R^n and Approximation theorems FD 3.22, Prop. 7.9, Theorem 7.10 Week10 Notes-Part A   03-16 Vitali-Caratheodory Theorem and Riesz Representation Theorem RC2.25, FD 7.2, Week10 Notes-Part B RC 6.18-6.19

Date Topics Sections Event Week 1 04-04 Fourier Transform I-Approximations, Theorem 8.14, Theorem 8.15 FD 8.2, EG 4.2.1, Week1 Notes-Part A Read FD Ch 8.1 , Ch8.2, in particular, Lemma/Prop. 8.4, 8.5-8.11. 04-06 Fourier Transformation II-Gaussian FD 8.3, LL 5.1-5.2, Week1 Notes-Part B Read FD Prop 8.17, 8.18, 8.23 Week 2 04-11 Fourier Transform III-Inversion FD 8.3 and LL 5.2-5.4, Week2 Notes-Part A Read LL 5.4 and FD Theorem 8.29 and Ch 8.7 04-13 Absolute continuous functions and Schwartz space FD 6.5, 8.4 and LL 5.5-5.8, Week2 Notes-Part B Read FD, Ch 8.6 Week 3 04-18 Distribution I-Definitions, Examples, and Operations LL 6.1-6.6, FD 9.1, Week3 Notes-Part A Read Page 285 of FD on translation and composition, Lemm/Prop 9.4 and 9.9 04-20 Distribution II-Derivatives, Convolution and Approximation LL 6.7-6.8, 6.12-6.13, FD 9.1, Week3 Notes-Part B Read FD, Ch 9.2, pages 291-294 for restricted distributions Week 4 04-25 Distributions III-Calculus with distributions LL 6.9-6.10, 6.22, Week4 Notes-Part A   04-27 Midterm exam Midterm exam This week's homework due time is extended for 48 hours Week 5 05-02 Two generalizations of Riesz-Markov Representation EG 1.8, LL 6.22, FD 7.1, Week5 Notes-Part A Compare the proof of Thm 7.2, FD with that of Thm 6.22, LL 05-04 The proof of Vector version RM FD 7.2, EG 1.8, Week5 Notes-Part B Read Prop/Thm 7.9 and 7.10 of FD Week 6 05-09 Distributions IV-Sobolev spaces LL 6.15-6.18, EG Ch 4.2   05-11 More Sobolev spaces LL 6.19-6.21, Week6 Notes   Week 7 05-16 Qualifying exam break Qualifying exam break   05-18 Qualifying exam break Qualifying exam break   Week 8 05-23 Rademacher's theorem EG 6.1, Week8 Notes-Part A   05-25 Convex functions EG 6.3, Week8 Notes-Part B   Week 9 05-30 Alexandrov's theorem on convex functions EG 6.4, Week9 Notes   06-01 Student Presentations Student Presentations   Week 10 06-06 Student Presentations Student Presentations   06-08 Student Presentations Student Presentations

Last modified: Wed April 14 7:06:11 PST 2022