Functional Analysis I

This course will explore how to deal with infinite dimensional vector spaces, particularly through Banach and Hilbert Spaces

Instructor: Prof. Chávez Casillas

Term: Spring

Course Overview

Functional analysis combines two fundamental branches of mathematics: Real Analysis and Linear Algebra. Basically, for an introductory course, you can expect Functional Analysis to be a “remake” of Linear Algebra on infinite dimensions. In addition, there are close connections between algebraic and topological properties in such spaces. Topics covered in the course sequence include: A review of fundamental concepts in Measure Theory, Normed Vector Spaces, Linear Operators and from there the course diverges to different application topics, specially geared towards PDEs.

Course Outcomes

This is a graduate-level theoretical mathematical course in Functional Analysis. The main outcomes of this course are:

  • Be able to work with fundamental concepts in functional analysis.
  • Understand and be able to apply the main, big theorems of functional analysis.
  • Be able to apply abstract ideas to concrete problems in analysis.
  • Understand the statements and proofs of the most important theorems.
  • Capacity to work with families of applications appearing in the course, particularly the ones geared towards the development of some ideas in PDEs.

Prerequisites

  • MTH 435 and MTH 436.

Textbooks

There is no official textbook for this class, but class notes will be provided on a regular basis. However, these notes were based on the following textbooks, which were the main source of information at some point.

  • Rynne, B. P., & Youngson, M. A. Linear Functional Analysis. Springer, 2008.
  • Moroşanu, G. Functional Analysis for the Applied Sciences. Springer, 2019.
  • Botelho, G., Pellegrino, D., and Teixeira, E. Introduction to Functional Analysis. Springer, 2025.
  • Arbogast, T., & Bona, J. L. Functional analysis for the applied mathematician. CRC Press, 2025.
  • Stein, E. M., & Shakarchi, R. Functional analysis: introduction to further topics in analysis (Vol. 4). Princeton University Press, 2011.
  • Khanfer, A. Applied Functional Analysis. Springer, 2024.
  • Kesavan, S. Functional analysis. Springer, 2023.

Grading

  • Midterm Exam 1: 20%
  • Midterm Exam 2: 20%
  • Homework: 40%
  • Cumulative Final Exam: 20%