Syllabus

Syllabus

Course description

There are many reasonable descriptions of functional analysis, but the way I like to think of it is: infinite dimensional linear algebra. It is linear algebra because it is concerned with various vector spaces, and it is infinite dimensional because, well, they are infinite dimensional vector spaces. This allows for the study of a much wider array of spaces than finite dimensional linear algebra because, for instance, even polynomials or spaces of continuous functions are infinite dimensional vector spaces. Moreover, since functional analysis at its core shares so much with linear algebra, you can imagine the vast applications it can have. So, you might be asking yourself: “where is the analysis” This is a valid question, and the answer lies in the word “infinite.” On our vector spaces, it will be important to discuss distances between vectors, or the length (norm) of vectors. We will prove during the course that all finite dimensional normed vector spaces of a given dimension are equivalent in a precise sense. This is far from true in the infinite dimensional case, and the richness this entails is part of what makes functional analysis so interesting.

We will use Introductory Functional Analysis with Applications, by Erwin Kreyszig. Prerequisites include the content of MATH-421 and MATH-450, especially topics related to abstract vector spaces, the dual of a vector space, inner product spaces, convergence, completeness, continuity and open and closed sets.

Course objectives

• Familiarity with a wide range of examples of metric, normed, Banach and Hilbert spaces.
• Ability to produce examples and counterexamples of illustrating topics in functional analysis.
• Solid understanding of the primary theorems in functional analysis (closed graph, open mapping, inverse mapping, principle of uniform boundedness, Hahn-Banach).
• Ability to prove the main theorems and apply them in novel situations.
• Recognize when to apply duality techniques to solve problems.

Course Outline And Topics

1. Metric spaces
2. Normed and Banach spaces
3. Inner product and Hilbert spaces
4. Fundamental theorems of Banach spaces
5. Spectral theory and compact operators
6. Functional calculi and the spectral theorem

Grading procedure

TBD

Important dates

Makeups

If an emergency arises which requires you to miss an exam, I must be made aware at least two hours prior to the start time of your exam. Note: proper documentation will be required before a makeup arrangement is considered.

Academic dishonesty/misconduct

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Disability Support Services

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Disclaimer

This syllabus is subject to change by the instructor if deemed necessary for the benefit of student learning or to correct errors and omissions.