Anastasios Matzavinos

Course Description

The content of Math 519/520 is focused on techniques of classical and functional analysis with applications to integral and partial differential equations. Topics in Math 519 include metric spaces, the basic theory of Hilbert and Banach spaces, Sobolev spaces, the contraction mapping theorem, distributions, Fourier series and Fourier transforms.

The ISU course catalog description for this course (including course requirements)  can be found here.

References

Grading Policy

The final grade will be based on homework assignments, a midterm exam (see below for dates) and one final exam, and it will be determined as follows:

Homework assignments Midterm Exam Final Exam

50% 20% 30%

Homework Assignments

Homework problems will be handed out by the instructor on a regular basis. Discussion of homework assignments with other students is encouraged, but what is handed in should be your own work. You are strongly encouraged to ask the instructor to go over those exercises that you find difficult.

The homework assignments and due dates are posted here.

Important Dates

Midterm exam Final Exam

Monday, October 19 Wednesday, December 16

Announcements box:

FRIDAY, DECEMBER 4: The eighth homework assignment has been posted online. MON, NOV 30 - WED, DEC 2: We defined the Schwartz space and introduced the concept of a tempered distribution. We also proved Plancherel's theorem for the Fourier transform. MON, NOV 23 - FRI, NOV 27: Thanksgiving break. MON, NOV 16 - FRI, NOV 20: We discussed weak solutions of PDEs and introduced the Fourier transform. Among other things, we proved the Fourier inversion theorem. FRIDAY, NOVEMBER 13: The seventh homework assignment has been posted online. WEDNESDAY, NOVEMBER 11: We focused on the convolution of distributions and proved that a distribution can be approximated by infinitely differentiable functions. MONDAY, NOVEMBER 9: Dr. Sacks discussed algebraic operations on distributions and defined the derivative of a distribution. Example 11 of Stakgold's chapter 3 was also covered. MON, NOV 2 - FRI, NOV 6: We defined local integrability and proved that the space of locally integrable functions is isomorphically embedded into the space of distributions. We focused on specific examples of distributions and defined principal value distributions. FRIDAY, OCTOBER 30: The sixth homework assignment has been posted online. MON, OCT 26 - FRI, OCT 30: We covered the basic concepts and theorems of Fourier analysis on the torus (chapter 7 of Hunter and Nachtergaele's book). FRIDAY, OCTOBER 23: We introduced the concepts of test functions and distributions. WEDNESDAY, OCTOBER 21: Dr. Yan proved theorem 7.3 on the density of trigonometric polynomials in the space of continuous functions on the torus. MONDAY, OCTOBER 19: The midterm exam will be held this Monday. WED, OCT 14 - FRI, OCT 16: We proved Bessel's inequality and turned our attention to the dual space of a Hilbert space. We stated and proved the Riesz representation theorem. MONDAY, OCTOBER 12: We defined unconditional convergence of unordered sums in Banach spaces and introduced the notion of an orthonormal basis for a Hilbert space. WED, OCT 7 - FRI, OCT 9: We discussed orthogonality in Hilbert spaces and focused on the existence and uniqueness of projections on closed linear subspaces of a Hilbert space. WEDNESDAY, OCTOBER 7: The fifth homework assignment has been posted online. MONDAY, OCTOBER 5: We introduced the dual space of a Banach space. We defined pre-Hilbert and Hilbert spaces and proved the Cauchy-Schwarz inequality. WED, SEP 30 - FRI, OCT 2: We introduced Banach spaces and Schauder bases, and we focused on the Banach space of bounded linear operators (sections 5.1 & 5.2). WEDNESDAY, SEPTEMBER 30: The fourth homework assignment has been posted online. MONDAY, SEPTEMBER 28: We proved the contraction mapping theorem (section 3.1) and discussed its applications to Fredholm integral equations (section 3.3). FRIDAY, SEPTEMBER 25: Dr. Sacks defined compact metric spaces and proved theorems 1.59 and 1.68. Also, the Arzela-Ascoli theorem on compact subsets of C(K) was introduced. WEDNESDAY, SEPTEMBER 23: We focused on complete metric spaces and proved that specific function spaces are complete. We also defined sequentially compact sets. MONDAY, SEPTEMBER 21: We have derived several useful inequalities, including those of Young, Holder and Minkowski. The third homework assignment has been posted online. WED, SEP 16 - FRI, SEP 18: We focused on linear and metric spaces. We introduced the notion of a Hamel basis for an infinite-dimensional linear space, we defined the Lebesgue spaces and proved Jensen's inequality for convex functions. MONDAY, SEPTEMBER 14: We defined the Lebesgue measure and integral. FRIDAY, SEPTEMBER 11: The second homework assignment has been posted online. WED, SEP 9 - FRI, SEP 11: We discussed the method of separation of variables in the context of the two-dimensional wave equation and studied the Dirichlet problem for Laplace's equation on a disc. We also introduced some elements of measure theory and defined the concepts of sigma-algebras and measurable spaces. MONDAY, SEPTEMBER 7: Labor Day - There will be no class on Monday. WED, SEP 2 - FRI, SEP 4: We continued our discussion of the method of characteristics in the context of first-order quasi-linear equations. We introduced the method of separation of variables and discussed D'Alembert's solution of the wave equation. MONDAY, AUGUST 31: We discussed the method of characteristics for first order linear PDEs. Also, the first homework assignment is now posted on the course web page. WED, AUG 26 - FRI, AUG 28: We introduced integral operators and linear integral equations. We derived solvability conditions for specific examples of Fredholm equations and showed that IVPs for linear ODEs are equivalent to integral equations of the Volterra type. MONDAY, AUGUST 24: We discussed the basic existence theory for ordinary differential equations and focused on initial and boundary value problems. MONDAY, AUGUST 24: Classes begin.

Disability Resources

Iowa State University complies with the Americans with Disabilities Act and Section 504 of the Rehabilitation Act. If you have a disability and require accommodations, please contact the instructor early in the semester so that your learning needs may be appropriately met. You should contact the Disability Resources office for information on appropriate policies and procedures. The Disability Resources office is located on the main floor of the Student Services Building, Room 1076; their phone is 515-294-6624.