MA199



1/22 - Organizational meeting, 12:10 - 1:00 in 959 Evans. Syllabus and other course details to be determined - we decided on one talk per course credit (together with attendance and participation) is sufficient for passing. For now, you can start reading about Hilbert spaces.

1/28 - Metric spaces - convergence, completeness, examples, normed and Banach spaces.

1/30 - Finish 1/28's lecture.

2/4 - Finish proof that bounded sequences are complete with respect to the supremum norm. Compute the completion of finitely supported sequences inside the set of bounded sequences.

2/6 - Continuity of a linear operator on a Banach space. The operator norm.

2/11 - Dual spaces and the Riesz representation theorem.

2/13 - Operators on Hilbert spaces form a C*-algebra.

2/18 - Holiday

2/20 - Wrap up infinite dimensional analysis.

2/25 - Start tensor products. and multilinear algebra, using the language of categories..

2/27 - More on categories and universal constructions; coproducts and products.

3/3 - Tensor products are unique up to a unique isomorphism; complexification as a tensor product. Indicate that V tensor W* is L(W,V), and that trace can be defined using this notion.

3/5 - Construction of bases for tensor products. Discussion of "when is a tensor product zero?" Identification of bilinear maps with matrices.



3/31, 4/2 - Alternating multilinear maps, the exterior algebra as a universal construction and as a quotient of the tensor algebra, and determinants as the realization of operators applied to the top exterior power.

4/7 - Begin modules over rings. Basic definitions and examples, and statement of the structure theorem for finitely generated modules over principal ideal domains.

4/9 - Indicate how the structure theorem for finitely generated abelian groups and the rational canonical form of a linear operator are special cases of the general structure theorem for modules over a PID.

4/14, 4/16, 4/21, 4/23 - The proper proof of the structure theorem, and proof that rank is well defined for modules over a commutative ring.

4/28, 4/30, 5/5 - Some facts about Banach algebras, leading up to the Gelfand representation and the characterization of commutative C* algebras.

5/7, 5/12 - An introduction to Galois theory, and a sketch of the fact that 5th and higher degree polynomials do not admit a general solution by radicals.