Math 7338 Functional Analysis


                                                                                                                       
                                                                                                                                    Instructor: Michael Loss
                                                                                         Lectures: TTh 12:05-13:25
                                                                                                                                     Location: Skiles 269

                                                                                       Office hours:   T,Th 11-12 , or by appointment 





                        Textbook:  Functional Analysis: An Introduction, by Yuri Eidelman, Vtali Milman and Antonis Tsolomitis
                                   Graduate Studies in Mathematics Vol 66, American Mathematical Society.  (EMT)

                   
                         Functional Analysis is a synthesis between linear algebra and analysis. It has its origins in the work of mathematicians like Vito Volterra (1860 - 1940)
                         and Eric Ivar Fredholm (1866-1929) on integral equations and was developed by the School around David Hilbert (1862 - 1943) and
                         especially by the School around Stefan Banach (1892 - 1945) and Hugo Steinhaus (1887 -1972).  Later, Functional Analysis turned out to be the key
                         language for formulating Quantum Mechanics in particular through the work of John von Neumann (1903-1957).

                         One of the key ideas in Functional Analysis is to consider functions as points in a space. While function spaces are the way to go in many applications, one can look at
                         common feature and proceed in an abstract fashion. This leads to notions like Banach spaces and Hilbert spaces. An important notion is completeness and
                         this notion allows to prove a number of general statements, like the uniform boundedness principle, the open mapping theorem and the closed graph theorem.

                         Functional analysis is also the starting point of fairly deep mathematical topics such as Banach Algebras and C*-algebras. I may make a few remarks about these.

                         A prerequisite of the course is some familiarity with notions like metric spaces, normed spaces,
                         open sets, closed sets, Cauchy Sequences, Completeness etc.. These are things that are taught in any undergraduate analysis course.

                         Here is a list of topics that I plan to cover in this course:

                         Normed spaces and linear operators, Hilbert spaces and linear operators on Hilbert spaces, Banach spaces and linear operators on Banach spaces,
                         the fundamental theorems: Hahn-Banach Theorem, Uniform Boundedness Theorem, Open Mapping Theorem and the Closed Graph Theorem,
                         Compact operators, Spectral theory of bounded self-adjoint operators and maybe an introduction to unbounded operators.

                         Here is a rough timeline with an approximate number of lectures devoted to each topic:

                         The Banach fixed point theorem and some of its applications, 3 lectures
                         Inner product spaces and Hilbert spaces: 3 lectures (Chapter 2 in EMT)
                         Duality: 3 lectures (Chapter 3 in EMT)
                         Bounded linear operators 4 lectures (Chapter 4 in EMT)
                         Spectral theory of compact operators 2 lectures (Chapter 5 in EMT)
                         Self-adjoint operators 4 lectures (Chapter 5 in EMT)
                         Spectral theory of self-adjoint operators (4 lectures)


                         Notes

                         HOMEWORK
                        

                        

                         Grades:   There will be homework which will count 100%
                                                  [100%, 90%] of the course work yields the grade A,
                                                 (90%, 80%] yields the grade B, (80%, 60%] yields the grade C, (60%, 50%] yields the grade D and below 50%
                                                 of the course work yields the grade F.