Math 637. Functional Analysis
- Fall 21 - Hans Lindblad

What the course is about: Functional Analysis is about Functionals, i.e. functions of functions. In particular Linear Functional Analysis extends Linear Algebra to spaces of functions, e.g. the spectral theorem, which in particular implies that a large class of functions can be expanded in a Fourier series, which in turn can be used for finding solutions to differential equations. There are a number of textbooks and lecture notes that start out well but at a certain point either become too abstract in their desire for generality or they focus on particular applications which do not display the general theory. I have chosen the classical book by Reed and Simon as our textbook because it avoids most of the abstraction while still proving a more general form of the spectral theorem. Still the other text books and lecture notes mentioned below explain things well in the beginning, and in particular the application to showing existence of solutions to PDEs is important, so I will complement this with material from the other books, e.g. Brezis and Taylor that have been texts for this course in previous years. Also, I do want to prove a more general version of the spectral theorem so I will complement with material from Simon's new book and from Hormander's lecture notes.

What the course will cover: Reed-Simon starts with a Chapter 'Preliminaries' with the real analysis that are prereq for this course such convergence of sequences of functions and the Lebesgue integral. The course will hence start with Chapter 2 about Hilbert spaces, in which you have an inner product. In Chapter 3 we will continue with the more general Banach spaces, in which you only have a norm. Chapter 4 is about Topology, but in too much generality for my taste so I think I will skip it and come back to it as needed. The main place it is used is in Chapter 5 about locally convex Topological Vector spaces, in particular Frechet Spaces, where you only have seminorms, e.g. the space of infinitely differentiable functions and their dual space of distributions. Chapter 6 is about bounded operators and their spectrum, in particular the spectral theorem for compact operators is proven. This in particular proves existence for Sturm Liouvillie operators and as a consequence convergence of Fourier series. Chapter 7 is about the Spectral theorem for bounded self adjoint operators. Chapter 8 is about the spectrum of unbounded operators. In addtion to this I hope to cover more about the Spectral Theorem from Simon and Hormander.

The structure of the course: The lectures are MW 1.30-2.45 in Hackermand 320. Please wear masks. I also hope that you will be working on the homework problems together. To help with this I and the grader Lili He will have an online problem session the Friday before they are due in place of office hours.

References:
  • Textbooks:
    • Reed and Simon 'Methods of Modern Mathematical Physics Vol 1: Functional Analysis' (revised and enlargened edition)
    Brezis 'Functional Analysis, Sobolev Spaces and Partial Differential Equations'
    Conway 'A course in Functional Analysis'
    Lax 'Functional Analysis'
  • Lecture Notes:
    • Melrose 'Introduction to Functional Analysis'
    Taylor 'Outline of Functional Analysis'
  • General Spectral Theorem:
    • Simon 'A comprehensive Course in Analysis Vol 4: Operator Theory'
    Hormander 'Linear Functional Analsyis'
  • Hilbert Spaces and Applications:
    • Debnaith and Mikusinski 'Introduction to Hilbert Spaces with Applications'
  • Distributions
    • Hormander 'The Analysis of Linear Partial Differential Operators Vol 1'
  • Nonlinear Functional Analysis
    • Zeidler 'Nonlinear Functional Analysis and its Applications Vol 1-4'
    Deimling 'Nonlinear Functional Analysis'
  • Reference works:
    • Yoshida 'Functional Analysis:
    Kato 'Perturbation theory for Linear Operators'
    Dunford and Schwarz 'Linear Operators Vol 1-3'