Math 7338 Functional Analysis


                                                                                                                       
                                                                                                                                    Instructor: Michael Loss
                                                                                         Lectures: TTh 9:35-10:55pm
                                                                                                                                     Location: Skiles 156

                                                                                       Office hours:   T,Th 12-1pm, or by appointment 





                        Textbook:  Introductory Functional Analysis with Applications, by Erwin Kreyszig, Wiley

                   
                         Functional Analysis is the 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). The power of functional analysis is due to the fact
                         that many problems like solving partial differential equations can be formulated in functional analytic terms. Such problems cannot be solved in terms
                         of a closed involving elementary functions. One always creates a sequence of functions and then shows that this sequence converges to the desired solution.
                         For this one needs a function space in which one expects to solution to be and a topology. Function spaces while linear spaces have the unpleasant property
                         that most of them are infinite dimensional. This is where the analysis comes in.

                         While function spaces are the way to go in many applications, one can look at common feature and proceed in an abstract fashion. The key notion here 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. Further, Functional Analysis gives a framework in which many problems in applied mathematics can be formulated. An example
                         is the Fredholm alternative for solving integral equations or the Neumann series.

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

                         A prerequisite of the course is analysis at least on the level of undergraduate analysis so that you are familiar with notions like metric spaces, normed spaces,
                         open sets, closed sets, Cauchy Sequences, Completeness etc..

                         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.

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

                         Chapter 2. Normed spaces and Banach spaces: 3 lectures,
                         Chapter 3. Inner product spaces and Hilbert spaces: 3 lectures
                         Chapter 4. Fundamental theorems for normed and Banach spaces: 5 lectures
                         Chapter 5. Banach fixed point theorem and its application to differential equations: 3 lectures
                         Chapter 7. Spectral theory of linear operators in normed spaces: 3 lectures
                         Chapter 8. Compact linear operators on normed spaces and their spectrum: 5 lectures
                         Chapter 9. Spectral theory of bounded self-adjoint linear operators: 5 lectures

                         Grades:   There will be homework and a final exam. The homework will count 70%
                                                 and the final exam 30%.   [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.


                        Homework 1, due Tuesday September 10.

                        Homework 2, due Thursday October 3.

                        Homework 3, due Thursday October 23

                        Homework 4, due Thursday November 14
                        
                        Homework 5, due Tuesday November 26

                        Final Exam