Math 273I
Numerical Functional Analysis
Lectures: TuTh 12:30-2:00pm, Room 55 Evans Hall
Office hours: TuTh 10:00-11:00am, Room 821 Evans Hall, and by appt.
Topics covered
- 08/28: Notation and basic terminology. Linear spaces and linear maps.
Kernel and range of a linear map. Quotient space.
- 09/02: Linear functionals, dual, bidual, column maps, construction of a basis. Dimension.
- 09/04: The interplay between column maps and row maps. Linear projectors. Duals of linear maps.
- 09/09: Applications: approximate evaluation of linear functionals and interpolation.
- 09/11: Topology defined. Continuity of maps. Metric spaces.
- 09/16: Modulus of continuity. Convergence of sequences. Contraction maps and fixed point iteration.
- 09/18: Compactness and total boundedness.
- 09/23: Normed linear spaces. Bounded linear maps.
- 09/25: Riesz' Lemma and consequences. Approximate inverses of linear maps.
- 09/30: The continuous dual. Representation of bounded linear functionals.
- 10/02: Application: interpolation error and optimal recovery.
- 10/07: Hahn-Banach Theorem, generalizations and consequences.
- 10/09: Pointwise convergence, w- and w*-convergence. Baire category theorem.
- 10/14: The uniform boundedness principle and consequences.
- 10/16: Open mapping and close graph theorems. Applications to ODEs.
- 10/21: Convexity. Topology of convex sets. Caratheodory's theorem.
- 10/23: The separation theorem. Best approximation from a convex set.
- 10/28: Best approximation from a linear subspace. Hahn-Banach for C(T). Chebyshev Alternation Theorem.
- 10/30: Krein-Milman Theorem. Inner product spaces: definition and basic properties of inner products.
- 11/04: Best approximations in Hilbert spaces. Riesz-Fischer representation theorem. Optimal interpolation.
- 11/06: Synge's hypercircle. Rayleigh-Ritz-Galerkin method.
- 11/13: Example: Poisson's equation. Lax-Milgram Lemma. Generalizations. Complete orthonormal systems.
- 11/18: Projection methods. Compact linear maps: definition and basic properties.
- 11/20: Compact perturbations of the identity. Fredholm Alternative.
- 11/25: Nystrom's method for Fredholm integral equations. Anselone's theorem and consequences.
- 12/02: Eigenstructure of bounded linear maps. Resolvent and spectral projectors. Spectral theorem for compact linear maps.
- 12/04: Linearization of nonlinear maps. Frechet and Gateaux derivatives. Meanvalue estimates.
- 12/09: Newton's method. Discretizations. Implicit function theorem via Newton.
The instructor welcomes cooperation among students and the use of books.
However, handing in homework that makes use of other people's work (be
it from a fellow student, a book or paper, or whatever) without
explicit acknowledgement is considered academic misconduct.
Homework is assigned every week, due at the beginning of class one
week later. A final 30-min presentation can be on any reasonably recent
research paper related to one of the topics of this class.
- Homework #1 due September 4th: Section 1, Problems 5, 7, 9, 11, 12, 13.
- Homework #2 due September 11th: Section 1, Problems 21, 31, 34, 37, 44, 47
- Homework #3 due September 18th: Section 2, Problems 2, 3, 8, 12, 14, 17.
- Homework #4 due September 25th: Section 2, Problems 22, 23, 26, 32. Section 3, Problems 5, 6.
- Homework #5 due October 2nd: Section 3, Problems 10, 13, 16, 17, 20. Section 4, Problem 1.
- Homework #6 due October 9th: Section 4, Problems 3, 4, 5, 8, 11, 12.
- Homework #7 due October 16th: Section 5, Problems 1, 4, 5, 7, 9, 12.
- Homework #8 due October 23rd: Section 5, Problems 19, 20, 23. Section 6, Problems 1, 2, 4.
- Homework #9 due October 30th: Section 6, Problems 8, 9, 10, 13, 14, 15.
- Homework #10 due November 6th: Section 6, Problems 17, 18, 21. Section 7, Problems 1, 3, 4.
- Homework #11 due November 13th: Section 7, Problems 6, 8, 10, 11, 15, 17.
- Homework #12 due November 20th: Section 7, Problems 18, 19, 20. Section 8, Problems 1, 2, 3.
- Homework #13 due December 2nd: Section 8, Problems 5, 7, 8. Section 9, Problems 2, 3, 4.
- Homework #14 due December 9th: Section 9, Problems 5, 6, 9.
Section 10, Problems 1, 2, 4.
Last modified: Dec 12, 2008