Information for Spring 2025 MAE 289C: Functional Analysis

Instructor: Prof. William McEneaney
- Office hour: T 4:50-5:50pm, our classroom or 1809 EBU1 (or just contact me to set up a time).
- Alternatively, feel free to bring up any questions regarding material homework and/or course plan in class.
- Office: 1809 EBU1.
- Email: wmceneaney@ucsd.edu

We can discuss grading ideas, but my nominal plan is as follows.
The grading would be based on the homework assignments and the take-home final.
To obtain your numerical score, you would add all your scores from the homework assignments and take-home final, and then divide that number by the total possible sum of those scores. If you like, you may multiply that by 100 to obtain a standard percentage score.
You must attend class at least 75% of the time in order to receive a passing grade.

H.L. Royden, Real Analysis (This contains a nice transition from real analysis to functional analysis.)
E. Kreyszig, Introductory Functional Analysis with Applications (This uses an unusual equivalent definition of many spaces, but is a very readable text.)
A.E. Taylor and D.C. Lay, Intro. to Functional Analysis (One of the classics of the field - a ``go to'' for me when I hit a tough issue.)
R.F. Curtain and H.J. Zwart, An Intro. to Infinite-Dimensional Linear Systems Theory, Appendices Only!!! (The appendices have a truly excellent discussion of func. analysis, and are particularly useful if we discuss closed linear operators at length, where these are useful for D&C problems with PDE dynamics.)
R. Adams, Sobolev Spaces (Chapter 3 (and some of the preceding material) is an excellent intro to Sobolev spaces, which we will likely use in some calculus of variations examples.)
There has been some discussion wrt covering Brownian motion (Wiener measure), SDEs (stochastic diffl. eqs.) and the Feynman/Kac and Fokker/Plank equations. In that case, some of the course may shift from classical functional analysis to the above probabilistic analysis, and the following references would replace some of the above ones.
B. Oksendal, Stochastic Differential Equations (A reasonably tractable discussion of SDEs and the more general context.)
I. Karatsas and S.E. Shreve, Brownian Motion and Stochastic Analysis (More extensive, with more detail than that in Oksendal.)

A relevant application of nonlinear functional analysis and calculus of variations can be found here.

L_p spaces
Normed vector spaces, Banach spaces
Space of continuous functions
Point-set topology in infinite-dimensional spaces (open, closed, compact, etc.)
Subspaces, some notions of basis
Bounded linear operators and functionals
Dual/conjugate spaces
Reisz representation theorems
Inner-product spaces, Hilbert spaces, some notions of basis
Hahn-Banach Theorem
Nonlinear functions, Gateaux and Frechet derivatives
Sobolev spaces
Calculus of Variations
Closed operators, closed graph theorem
Integral equations, Fredholm operators
Spectral analysis, approximation
IF we decide to make a side quest into stochastic processes, some of the above topics would be replaced with the following.
Brownian motion and Wiener measure
Ito integrals
Stochastic differential equations and notions of solution
Girsanov transform
Feynman-Kac and Fokker-Planck