Special Mathematics Lecture (Introduction to functional analysis)

Goals of the Course

Mathematics play a central role in all natural and social sciences. It is a universal language which can be learned, enjoyed, and shared by everybody. The goal of this course is to provide a solid knowledge on various mathematical topics. Each semester, a new subject is proposed, and the presentation is accessible to all motivated students, independently of their major and of their academic age. These lectures also offer a unique opportunity to study in an international, transdisciplinary, and friendly environment.

Objectives of the Course

Functional analysis is a useful tool for many physical theories, and has been partially developed concomitantly with quantum mechanics. The objective of this course is to provide the necessary background for a good understanding of the mathematics behind any course of quantum mechanics. During this one semester course, the notions of distributions, of Lebesgue integral, and the foundation of spectral theory will be introduced. Depending on the interest of the audience, different tools of spectral theory will be further developed.

Course Contents or Plan

This course should cover the following topics:

1. Distribution theory
2. Lebesgue theory of integration
3. Operator theory on Hilbert spaces

Course Prerequisites and Related Courses

Basic knowledge on calculus and linear algebra, as provided in Calculus I & II and in Linear algebra I & II. Motivated 1st year students can also attend without these prerequisites but after a discussion with the instructor.

Course Evaluation Method and Criteria

The final grade will be based on the active participation during the lectures and on some written reports. Students will be encouraged to work on applications related to their major during the semester. Students need to notify the course withdrawal to the instructor when they have no intention of finishing the course during the semester.

Study Load (Self-directed Learning Outside Course Hours)

Students are expected to read their notes, and to be familiar with the content of the previous lectures before each new lecture.

Notice for students

It is expected that the students will show a certain maturity in studying independently and in choosing some exercises and problems to solve. Study sessions will be organized on a weekly basis. For students enrolled in or after AY 2022, this course is a Global Liberal Arts course and can be counted towards graduation. For students enrolled in or before AY 2021, this course is an optional course in any program at Nagoya University.

Textbook

Lecture Notes

Additional information

Reference Website

This lecture is provided under Creative Commons Attribution-Non Commercial-ShareAlike 4.0 International.

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