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In 2016, the topic of this course is FUNCTIONAL ANALYSIS.

Syllabus

In general, functional analysis can be defined as the study of infinite dimensional vector spaces and operators on these spaces. Functional analysis as a separate mathematical subject emerged about a century ago, motivated in part by the success of using Hilbert spaces to rigorously justify Fourier series expansions for functions on the unit circle. In the extension of this theory to other settings, fundamental questions in functional analysis appear: (i) What are the appropriate topologies to put on the infinite dimensional vector spaces that appear as spaces of functions? (ii) To what extent do linear differential operators acting on spaces of functions (like d/d theta acting on functions on the circle) behave like linear transformations on finite dimensional vector spaces?

The first question is highly nontrivial and depends on the context, precisely because infinite dimensional vector spaces have many inequivalent norm topologies. Investigating the second question quickly leads to the realization that even the simplest linear differential operator like d/d theta is discontinuous in the most reasonable topologies. On the other hand, standard operators like Green's operators (e.g. (d/d theta)^{-2}, roughly speaking the inverse of the Laplacian on the circle) are continuous in these topologies. Therefore, functional analysis naturally splits into the study of continuous operators on function spaces and discontinuous operators - both are important.

More specifically, this course will cover the basic topologies on infinite dimensional vector spaces, including Hilbert spaces (inner product topologies), Banach spaces (norm topologies), and Frechet spaces (topologies built to handle spaces of smooth functions). Specific examples will include L^p and Sobolev spaces. We will restrict attention to continuous linear operators between these spaces, with applications to the study of differential operators. We will discuss the spectral theorem about diagonalization of linear operators on Hilbert spaces with Fourier series as the guiding example. As time permits, we'll discuss more advanced results like the Hahn-Banach theorem and the theory of distributions, the rigorous treatment of delta functions.

Learning Outcomes

On successful completion of this course, students will be able to

1. Define complete normed spaces, and understand the basic examples of Banach and Hilbert spaces;
2. Define Frechet spaces and understand the basic examples;
3. Understand the spectral theorem for compact selfadjoint operators on Hilbert spaces, with basic examples;
4. Understand the Hahn-Banach theorem, open mapping theorem and closed graph theorem;
5. Understand the basics of distribution theory.

Assumed knowledge: Topology and Analysis III.

None.

1. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations"

2. Conway, "A Course in Functional Analysis"

3. Kreyszig, "Functional Analysis With Applications"

4. Riesz and Nagy, "Functional Analysis"

5. Rudin, "Functional Analysis"

The course will have an active MyUni website.

The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

An aggregate score of 50% is required to pass the course.

There will be a total of 6 homework assignments, due one week after assigned. Each will cover material from the lectures, and in addition, will sometimes go beyond that so that students may have to undertake some additional research.

Homework assignments must be given to the lecturer in person or emailed as a pdf. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.