- ACM105
ACM ACM ACM

ACM 105
Applied Real and Functional Analysis

Instructor
Kathryn Leonard
Office: Firestone 315
Phone: 626.395.4550
Office Hours: after class or by appt.
Website: www.acm.caltech.edu/~kathryn
Email: kathryn_at_acm_dot_caltech_dot_edu

TA
Patrick Dondl
Office: Thomas 320
Office Hours: Monday 1-3 PM
Email: pwd_at_caltech_dot_edu

Topics
The Lebesgue integral on the line, general measure and integration theory, convergence theorems, Fubini, Tonelli, the Lebesgue integral in n dimensions and the transformation theorem, L_p spaces, convolution, Fourier transform and Sobolev spaces with application to PDEs, the convolution theorem, Friedrich's mollifiers, dense subspaces and approximation, normed vector spaces, completeness, Banach spaces, linear operators, the Baire, Banach-Steinhaus, open mapping and closed graph theorems with applications to differential and integral equations, dual spaces, weak convergence and weak solvability theory of boundary value problems, spectral theory of compact operators.

Grading
There will be homework assignments throughout the semester as well as a take-home final and an in-class presentation. Grades will be assigned roughly according to the following allocation, although instructor discretion will play some role.
HW: 60%
Presentation: 20%
Final: 20%.

Date  Chapter  Topic  Comments & Homework 
       
9/25  1.1, 1.2 Metric spaces  1.1 #4, 8; 1.2 #4, 8, 11
9/27  1.3, 1.4, 1.5 Structure of metric spaces  1.3 #6, 12; 1.4 #5, 6, 8; 1.5 #3, 8, 11, 12
10/1  1.6 Completion  1.6 #4,6,12. DUE TUESDAY, 10/12: all work from 9/25-10/1. Turn into ACM 105 box in Firestone lobby.
       
10/4  no class    
10/6  2.1-2.2  Normed vector spaces, measures  
10/8     More measure  HW 2 is here . Due Tuesday, Oct. 19.
       
10/11     Measurable functions  
10/13    Simple functions and Lebesgue integral;  
10/15     Properties of Lebesgue integral  HW 3 is here . Due Tuesday, Oct. 26. (hereafter HW will be due the following Friday.)
       
10/18    Convergence and integrals  
10/20    L^2, L^p  
10/22    Back to normed spaces...  HW 4 is here . Due Friday, Oct. 29. Also due Friday is a project description (but turn that in to me).
       
10/25  2.5, 2.6 Linear operators  2.5 #7; 2.6 #14, 15
10/27  2.7, 2.8 Linear functionals   2.7 #2, 10; 2.8 #6, 15
10/29  2.9, 2.10 Dual spaces   2.9 #10, 13; 2.10 #8, 10. HW is due Friday, Nov. 5.
       
11/1  3.1, 3.2 Hilbert spaces  3.1 #11, 15; 3.2 #4, 10
11/3  3.3, 3.4 Orthonormal sets and sequences   3.3 #3; 3.4 #8, 9
11/5  3.6, 3.7 Total sets   3.6 #8,9; 3.7 #1, 3. HW is due Friday, Nov. 12.
       
11/8  3.8, 3.9, 3.10 Hilbert-adjoint operators  
11/10  4.1, 4.2 Hahn-Banach Theorem  
11/12  4.3, 4.4 H-B Thms. and applications   3.8 #14; 3.9 #3; 3.10 #15; 4.3 #8, 11. HW is due Friday, Nov. 19. Start thinking about your projects!!
       
11/15  4.4, 4.5 Dual of C([a,b]), Adjoint operators  
11/17  4.6, 4.7 Reflexive spaces, Baire Category Thm, consequences  
11/19  4.8, 4.9, 4.12 Open Mapping Thm, weak, weak* convergence   No HW this week--work on projects instead.
       
11/22  4.10 Summability  
11/24  4.11 Numerical integration  
11/26   Break!  
       
11/29  Paige, Chung-Yang Wavelets  
12/1  Taiala Finite element method  
12/3  Luis, Peter Wavelets, learning theory   Papers are due today by midnight. Final exam is here. Due Wendesday, Dec. 8 by midnight.
       

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