Introduction to Functional Analysis, with ApplicationsT/Th 2-3:20105 EBU 2

Instructor: Prof. William M. McEneaneyOffice: 1209 EBU 1Phone: X2-5835Email: wmceneaney@ucsd.edu

Office hours: W 3–4 (or contact me to set up a time).

Text: Kreyszig, Introductory Functional Analysis with Applications.

Additional books providing other perspectives on the material in the text-book and covering topics from nonlinear functional analysis may be placedon reserve in the library.

Content: The language and concepts of Functional Analysis are ubiquitous in modernEngineering Theory. The purpose of this course is to rigorously introducethe fundamentals of Functional Analysis in a slightly condensed form. Al-though much of the material may seem rather abstract, rest assured thatthe presentation is from a more practical perspective than what is the casefor pure Mathematics majors. At the end of the course, you will be able towork with objects in Banach and Hilbert spaces (e.g., L2), and some of themost relevant tools and results.Although the definitions of many useful spaces typically involve Lebesgueintegration, we will study the underlying structure without the machineryof Lebesgue integration. This hugely reduces the upfront costs of learningthe material, with only very small disadvantages from our perspective. Thespaces of interest are infinite dimensional, and may be spaces of sequencesor functions of one or many variables. We will define a number of spaces,and explore the properties of several classes of such spaces. We will thenexamine the (linear) functions on these spaces and the structure of sets ofthese functions.We will attempt to include some material on applications. This could in-clude the use of fixed point theory to demonstrate the existence of solutions of(ordinary and partial) differential and integral equations, weak convergenceconcepts and their application to proofs of convergence of numerical methodsfor differential equations, and the Calculus of Variations. In this last appli-cation area, we would examine differentiation on infinite-dimensional spaces,the extension of Lagrange multipliers to that domain, and the PontryaginMaximum Principle in Optimal Control.

Grading: (Obviously TBD!)Homework: ?%Quiz or not?: ?%Presentation or not?: ?%Take-Home Exam: ?%

(Note: Homework is due at the beginning of class on the due date. Home-work handed in later that day will receive a 10% deduction. Homeworkcannot be handed in after the due date.)