18.102 | Spring 2021 | Undergraduate

# Introduction to Functional Analysis

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.

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

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

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

Lecture 10: Simple Functions (PDF)

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

### Week 7

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 Lp spaces
• The completeness of Lp 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)

• Cesaro-Fourier means and the Fejer kernel
• Convergence of the Cesaro-Fourier means of an L2 function
• The normalized exponentials form an orthonormal basis for L2

### Week 10

Reading: [RM] Chapter 3, Sections 8–10

Lecture 17: Minimizers, Orthogonal Complements and the Riesz Representation Theorem (PDF)

Lecture 17: Minimizers, Orthogonal Complements and the Riesz Representation Theorem (TEX)

• Length minimizers for closed convex sets
• Orthogonal complements
• The Riesz representation theorem

### Week 11

Reading: [RM] Chapter 3, Sections 10–14

Lecture 18: The Adjoint of a Bounded Linear Operator on a Hilbert Space (PDF)

Lecture 18: The Adjoint of a Bounded Linear Operator on a Hilbert Space (TEX)

• Adjoints of bounded linear operators
• Compact sets and equi-small tails

Lecture 19: Compact Subsets of a Hilbert Space and Finite-Rank Operators (PDF)

Lecture 19: Compact Subsets of a Hilbert Space and Finite-Rank Operators (TEX)

• Characterization of compact subsets of Hilbert spaces
• Finite rank operators
• The definition of compact operators

### Week 12

Reading: [RM] Chapter 3, Sections 14, 16–18

Lecture 20: Compact Operators and the Spectrum of a Bounded Linear Operator on a Hilbert Space (PDF)

Lecture 20: Compact Operators and the Spectrum of a Bounded Linear Operator on a Hilbert Space (TEX)

• Compact operators and their properties
• The resolvent set and spectrum of a bounded linear operator

Lecture 21: The Spectrum of Self-Adjoint Operators and the Eigenspaces of Compact Self-Adjoint Operators (PDF)

Lecture 21: The Spectrum of Self-Adjoint Operators and the Eigenspaces of Compact Self-Adjoint Operators (TEX)

• The spectrum for a self-adjoint operator
• The eigenvalues and eigenspaces of compact self-adjoint operators

### Week 13

Readings: [RM] Chapter 3, Section 18 and Chapter 4, Section 4

Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator (PDF)

Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator (TEX)

• The maximum principle for finding eigenvalues and eigenvectors
• The spectral theorem for compact self-adjoint operators

Lecture 23: The Dirichlet Problem on an Interval (PDF)

Lecture 23: The Dirichlet Problem on an Interval (TEX)

• Well-posedness for the Dirichlet problem on an interval

Spring 2021
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