Lecture Notes and Readings

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 MIT student Andrew Lin who took the class with Dr. Rodriguez in 2021. (Both sets of notes used with permission.)

[RM] = Functional Analysis (PDF - 1.2MB) lecture notes by Richard Melrose, Spring 2020

Dr. Rodriguez’s Fall 2021 lecture notes in one file.

Introduction to Functional Analysis (PDF)
Introduction to Functional Analysis (ZIP) LaTeX source files

Individual lecture notes files are below.

Week 1

Reading: [RM] Chapter 1, Sections 1–4

Lecture 1: Basic Banach Space Theory (PDF)

Lecture 1: Basic Banach Space Theory (TEX)

Finite dimensional and infinite dimensional vector spaces
Normed spaces and Banach spaces
Examples of Banach spaces including little lp spaces and the space of bounded continuous functions on a metric space

Lecture 2: Bounded Linear Operators (PDF)

Lecture 2: Bounded Linear Operators (TEX)

An equivalent condition, in terms of absolutely summable series, for a normed space to be a Banach space
Linear operators and bounded (i.e. continuous) linear operators
The normed space of bounded linear operators and the dual space

Week 2

Reading: [RM] Chapter 1, Sections 5–12

Lecture 3: Quotient Spaces, the Baire Category Theorem and the Uniform Boundedness Theorem (PDF)

Lecture 3: Quotient Spaces, the Baire Category Theorem and the Uniform Boundedness Theorem (TEX)

Subspaces, quotients, and obtaining a normed space from a semi-normed space
The Baire category theorem
The uniform boundedness theorem

Lecture 4: The Open Mapping Theorem and the Closed Graph Theorem (PDF)

Lecture 4: The Open Mapping Theorem and the Closed Graph Theorem (TEX)

The open mapping theorem
The closed graph theorem
Zorn’s lemma (in anticipation of the Hahn-Banach theorem)

Week 3

Reading: [RM] Chapter 1, Sections 12–13

Lecture 5: Zorn’s Lemma and the Hahn-Banach Theorem (PDF)

Lecture 5: Zorn’s Lemma and the Hahn-Banach Theorem (TEX)

The proof that every vector space has a Hamel basis using Zorn’s lemma
The proof (- epsilon) of the Hahn-Banach theorem

Lecture 6: The Double Dual and the Outer Measure of a Subset of Real Numbers (PDF)

Lecture 6: The Double Dual and the Outer Measure of a Subset of Real Numbers (TEX)

The double dual of a normed space
The need for integration better than Riemann
Towards Lebesgue measure, the definition and elementary properties of outer measure

Week 4

Reading: No readings

Lecture 7: Sigma Algebras (PDF)

Lecture 7: Sigma Algebras (TEX)

The definition of a Lebesgue measurable set
Algebras and sigma algebras of subsets of real numbers
The Borel sigma-algebra

Week 5

Reading: No readings

Lecture 8: Lebesgue Measurable Subsets and Measure (PDF)

Lecture 8: Lebesgue Measurable Subsets and Measure (TEX)

The collection of Lebesgue measurable sets is a sigma-algebra containing the Borel sigma-algebra
The definition and properties of Lebesgue measure

Lecture 9: Lebesgue Measurable Functions (PDF)

Lecture 9: Lebesgue Measurable Functions (TEX)

Motivation for the definition of Lebesgue measurable functions
Properties of extended real-valued, Lebesgue measurable functions

Week 6

Reading: No readings

Lecture 10: Simple Functions (PDF)

Lecture 10: Simple Functions (TEX)

Complex-valued measurable functions
Approximation of measurable functions by simple functions
The definition of the Lebesgue integral for nonnegative simple functions

Week 7

Reading: No readings

Lecture 11: The Lebesgue Integral of a Nonnegative Function and Convergence Theorems (PDF)

Lecture 11: The Lebesgue Integral of a Nonnegative Function and Convergence Theorems (TEX)

The definition of the Lebesgue integral of a nonnegative measurable function
The monotone convergence theorem and consequences
Fatou’s lemma

Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence Theorem (PDF)

Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence Theorem (TEX)

The definition of a Lebesgue integrable function and its integral
The dominated convergence theorem
The Lebesgue integral of a continuous function equals its Riemann integral

Week 8

Reading: [RM] Chapter 3, Sections 1–4

Lecture 13: L{{< sup “p” >}} Space Theory (PDF)

Lecture 13: L{{< sup “p” >}} Space Theory (TEX)

The definition of L^p spaces
The completeness of L^p spaces
Pre-Hilbert spaces and the Cauchy-Schwarz inequality

Lecture 14: Basic Hilbert Space Theory (PDF)

Lecture 14: Basic Hilbert Space Theory (TEX)

The norm induced by the inner product on a pre-Hilbert space
Hilbert spaces
Orthonormal and maximal orthonormal subsets of a pre-Hilbert space

Week 9

Readings: [RM] Chapter 3, Sections 5–6 and Chapter 4, Section 1

Lecture 15: Orthonormal Bases and Fourier Series (PDF)

Lecture 15: Orthonormal Bases and Fourier Series (TEX)

Orthonormal bases of a Hilbert space
Classification of separable, infinite dimensional Hilbert spaces (there’s only one)
Fourier series of an L2 function

Lecture 16: Fejer’s Theorem and Convergence of Fourier Series (PDF)

Lecture 16: Fejer’s Theorem and Convergence of Fourier Series (TEX) fejer.PNG