Introduction to Functional Analysis
Course facts
- Payment: Free, no certificate
- Credits: Non Credit
- Learning style: Self-paced, start anytime
- Resources: Editable Files; Exams; Lecture Notes; Lecture Videos; Problem Sets
MIT OpenCourseWare details
- As taught in: Spring 2021
- Department: Mathematics
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Video course on Banach spaces, Lebesgue integration, Hilbert spaces and the Spectral Theorem, with lecture notes, ten assignments and take-home exams.
When a problem lives on an infinite-dimensional space, ordinary linear algebra runs out. Functional analysis extends it, as in describing a confined quantum particle’s state as a superposition of elementary wave functions. This class builds the theory through video lectures, starting with normed spaces and integration, then reaching the Spectral Theorem.
What You Will Achieve
- Prove core results on Banach spaces such as the Hahn-Banach Theorem
- Construct Lebesgue measure and the Lebesgue integral
- Use orthonormal bases and Fourier series in Hilbert spaces
- Apply the Spectral Theorem for compact self-adjoint operators
Program Overview
Banach Spaces
The opening lectures set out normed spaces and their big theorems.
- Bounded linear operators
- The Baire Category Theorem
- The Open Mapping and Closed Graph Theorems
- Zorn’s Lemma and the Hahn-Banach Theorem
Lebesgue Integration
A new integral is needed for complete function spaces.
- Sigma algebras
- Lebesgue measurable functions
- The Dominated Convergence Theorem
- L p space theory
Hilbert Spaces and Operators
The final part brings the two threads together.
- Orthonormal bases and Fourier series
- The Riesz Representation Theorem
- Compact operators
- The Spectral Theorem
Course Completion
Job Outlook
This is core preparation for graduate mathematics, and it supports work in partial differential equations and mathematical physics.
Good for: Mathematics students who have finished real analysis and linear algebra and enjoy proofs.
Keep in mind: From spring 2021, with no assigned textbook; you follow Prof. Melrose’s notes and a student’s notes. Assignments and exams are included, but the materials list no solutions, so you must check your own proofs.
Bottom line: A clean, complete video course on a core subject, and a natural next step after real analysis.
Frequently Asked Questions
Is there a textbook?
No. The class follows notes Richard Melrose wrote when he taught it in 2020, along with notes taken by a student in the 2021 class.
How were exams given?
As take-home work: a 24-hour midterm in week seven and a 48-hour final assignment at the end, both of which are published.
Where does functional analysis show up in practice?
The description uses a nonrelativistic quantum particle as its example: its state can be written as a superposition of wave functions with increasing energies.
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