Functional Analysis

A brief history of the subject:

Linear Functional Analysis is mostly concerned with the study of continuous linear maps between topological vector spaces and as such it comprises an inﬁnite dimensional analogue of linear algebra. It is only through a ﬁne and delicate mixture of linearity and continuity (analysis) that such a miraculous foray into the inﬁnite dimensional realm is possible. Some of the most important classes of topological vector spaces, with increasing order of generality, include: Hilbert spaces, Banach spaces, Frechet spaces and locally convex spaces. Integration and diﬀerentiation provide some of the most important classes of linear maps between spaces of functions in the form of integral operators and partial diﬀerential operators.

In fact the original impetus for the development of functional analysis came from the work of Volterra and Fredholm on calculus of variations and integral operators, respectively. Motivated by these works, Hilbert and his school, in the ﬁrst decade of the 20th century, established the ﬁrst general results of functional analysis, most notably, the spectral theorem of compact operators on an inﬁnite dimensional Hilbert space.

After these early developments, a second major motivation for further development of functional analysis came from the mathematical structures hidden in quantum mechanics, as originally developed by Heisenberg and Schroedinger in mid 1920’s. In the hands of Dirac, and in a more rigorous way von Neumann (who in fact deﬁned abstrcat Hilbert spaces and coined the term Hilbert space), Hilbert space and its algebra of operators became the main playing ground for formulating the postulates of quantum mechanics. Partial diﬀerential equations and analysis in general are other main users of functional analysis. In fact it is hard to think of any branch of modern analysis, geometry, number theory, or topology that is not aﬀected in serious ways by techniques of functional analysis. Finally, the very active area of operator algebras (most notably C* -algebras and von Neumann algebras) is one of the latest outgrowths of functional analysis. These two subjects are the backbone of noncommutative geometry.

While our main emphasis in this course is the spectral theory of bounded operators on a Hilbert space, several foundational results will be proved in the more general context of Banach spaces.

Outlines and Lecture notes:

• Geometry of Hilbert space, Riesz representation theorem, Hilbert basis; an example from ergodic theory,
• Banach spaces and the three pillars of functional analysis: Hahn-Banach the- orem, closed graph theorem, open mapping principle,
• The spectral theorem for compact self adjoint operators, applications: Peter-
• Weyl theorem, boundary value problems and Sturm Liouville theory,
• Topologies on bounded operators, trace class, and Hilbert Schmidt operators,
• The spectral theorem for bounded normal operators on Hilbert space,
• Introduction to unbounded operators, Stone’s theorem.
You can find pdf file of the topics Discussed per each class here.

Marking Policy:  50% assignments, 30% midterm exam, 20% project. Projects will be chosen in consultation with each student, and usually reflect student’s area of specialization or interests. I expect students to prepare a short essay on their project and present it in class in a 50 minutes lecture towards the end of the term. Students should make sure to see me no later than 3 weeks after the start of the class regarding their projects.

Prerequisites:

linear algebra, general topology, undergraduate analysis, and some measure theory will be useful.

Assignments:

• Problems Set 1: PDF
• Problems Set 2: PDF

Textbook:

 I shall closely follow Reed and Simon’s Functional Analysis, I, from Chapter 2 (Hilbert spaces), and skip Chapters 1 and 4 which are just background material. Complementary material will be presented from other sources.

Students' Presentation Notes

Disclaimer::  these note, on selected topics, are solely written by students in the course. Ideally, I should have spent time with each student helping her/him with the text and editing it. Well, hopefully next time! I made many comments during the presentations though....

Here are the topics and notes: