18.102 | Spring 2021 | Undergraduate

# Introduction to Functional Analysis

## Syllabus

### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Prerequisites

18.06 Linear Algebra, 18.700 Linear Algebra, or 18.701 Algebra I; and 18.100A Introduction to Analysis, 18.100B Analysis I, 18.100P Real Analysis, or 18.100Q Real Analysis; or permission of the instructor.

### Description

Functional analysis helps us study and solve both linear and nonlinear problems posed on a normed space that is no longer finite-dimensional, a situation that arises very naturally in many concrete problems. For example, a nonrelativistic quantum particle confined to a region in space can be modeled using a complex valued function (a wave function), an infinite dimensional object (the function’s value is required for each of the infinitely many points in the region).  Functional analysis yields the mathematically and physically interesting fact that the (time independent) state of the particle can always be described as a (possibly infinite) superposition of elementary wave functions (bound states) that form a discrete set and can be ordered to have increasing energies tending to infinity.

The fundamental topics from functional analysis covered in this course include normed spaces, completeness, functionals, the Hahn-Banach Theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of Lᵖ spaces; Hilbert spaces; compact and self-adjoint operators; and the Spectral Theorem.

### Textbooks and Notes

There is no assigned textbook for this course. Instead, we will follow lecture notes written by Professor Richard Melrose when he taught the course in 2020, as well as lecture notes taken by an MIT student who took the class with Dr. Rodriguez in 2021.

These can be found in the Lecture Notes and Readings section.

### Assignments

#### Weekly Homework

The written homework is extremely important (mathematics is not a spectator sport). The best way to test your knowledge of a concept is to try and use it; this is why you work problems. Collaboration with other students in the class is encouraged, but separate solutions must be written up and collaborators documented at the top right-hand corner of all submitted work. In general, late homework will not be accepted. However, circumstances may arise that warrant an extension; such an extension request should be emailed to the instructor. The lowest individual homework grade will be dropped when computing the final homework grade for the course.

#### Midterm and Final Assignment

The midterm exam is a 24-hour take-home exam taking place in week 7. The final assignment will be a 48-hour assignment at the end of the course, which looks and smells like an exam but is not one due to technical considerations.

activities percentages
Weekly Homework 50%
Midterm Exam 25%
Final Assignment 25%

Final letter grade cutoffs will be less than or equal to the “standard cutoffs,” e.g. 90 for A, 80 for B, etc.

Spring 2021
Problem Sets
Exams
Lecture Notes
Lecture Videos