Simon Fraser University
Engaging the World

Course Outlines

Spring 2026 - APMA 905 G100

Applied Functional Analysis (4)

Class Number: 5253

Delivery Method: In Person

Overview

  • Course Times + Location:

    Jan 5 – Apr 10, 2026: Wed, Fri, 10:30 a.m.–12:20 p.m.
    Burnaby

Description

CALENDAR DESCRIPTION:

Infinite dimensional vector spaces, convergence, generalized Fourier series. Operator Theory; the Fredholm alternative. Application to integral equations and Sturm-Liouville systems. Spectral theory.

COURSE DETAILS:

Course description:
This will be an introduction to functional analysis and some of its applications. In a nutshell, we'll investigate the properties of continuous linear mappings of infinite-dimensional vector spaces. The more structure the space has, the stronger the theorems. We will focus on Hilbert spaces (which are almost 'tame') and Banach spaces (which can be wilder). The results will provide extremely useful tools for applications ranging from fixed point theory to numerical approximation through to the solvability of partial differential equations. We could also sell this course as the 'fundamental underpinning of much of machine learning', but we won't.

Topics covered: We'll quickly review important definitions concerning metric spaces and normed spaces, especially the notions of completeness and compactness. We'll then introduce Banach spaces and bounded linear operators on them. We shall then discuss Hilbert spaces and their duals, some applications, and then onto spectral theory/Sturm-Liouville theory. We will pause to marvel at the simple yet far-reaching Riesz Representation theorem and the Lax-Milgram Lemma. We then return to abstract Banach spaces, and the 'big' theorems: the Hahn-Banach theorem, the Uniform Boundedness theorem, the Open Mapping theorem and the Closed Graph theorem. The course ends with a return to spectral theory, and the Fredholm Alternative for compact operators.

Structure and Assessment: There will be 4 hours of lecture, and 1 hour of problem session (for a total of 5 hours of contact time) per week. There will be 12 equally-weighted weekly assignments. Students will present their solutions in the problem sessions, have an opportunity to correct their own work, and the assessment will be based on the oral and written solutions. Note that we may, based on the success of these problem sessions, change the method of assessment to include a final examinatiom.

The work you hand in must represent your own efforts. Plagiarism is a serious violation of SFU'S Academic Honesty and Student Conduct policy, and we maintain a zero-tolerance approach to such violations.

Grading

  • 12 Assignments (equally weighted) 100%
  • Final exam (depending on the success of the assignments) 0%

REQUIREMENTS:

Prerequisites:
A good (advanced undergraduate/beginning graduate) background in linear algebra  and real analysis. You should be comfortable with notions of linear spaces, linear independence, eigenvalues/eigenfunctions, and metric spaces. Measure theory and graduate PDE would be useful courses to have had, but are not mandatory prerequisites.

Materials

RECOMMENDED READING:

INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS,

KREYSZIG, E., WILEY & SONS, INCORPORATED.

Please note that this textbook is not required.


ISBN: 9780471507314

REQUIRED READING NOTES:

Your personalized Course Material list, including digital and physical textbooks, are available through the SFU Bookstore website by simply entering your Computing ID at: shop.sfu.ca/course-materials/my-personalized-course-materials.

Graduate Studies Notes:

Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.

Registrar Notes:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

At SFU, you are expected to act honestly and responsibly in all your academic work. Cheating, plagiarism, or any other form of academic dishonesty harms your own learning, undermines the efforts of your classmates who pursue their studies honestly, and goes against the core values of the university.

To learn more about the academic disciplinary process and relevant academic supports, visit: 


RELIGIOUS ACCOMMODATION

Students with a faith background who may need accommodations during the term are encouraged to assess their needs as soon as possible and review the Multifaith religious accommodations website. The page outlines ways they begin working toward an accommodation and ensure solutions can be reached in a timely fashion.