Office hours: TuTh 10:00-11:00am, Room 821 Evans Hall, and by appt.

- 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.

- Lecture notes by Carl de Boor.
- Additional reading also suggested by Carl de Boor.

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