Lectures

Week 3

We cover Section 7.1 and 7.2 in Spaces as one combined lecture.

Lecture 1 & 2: Measure spaces

• 1.1 Introduction
• 1.2 Sigma algebras
• 1.3 Measure spaces
• 1.4 Null sets
• 1.5 Complete measure spaces
• 1.6 Lebesgue measure

Week 4

We cover Section 7.3 (Lecture 3) and 7.4 (Lecture 4) in Spaces. It might be useful to take a look at Section 1.4 in Spaces before watching the videos. 

Lecture 3: Measurable functions

• 3.1 Introduction
• 3.2 Basic properties
• 3.3 Combinations
• 3.4 Limits

Remark. Lemma 11 should also include the following statement: "Conversely, if f is measurable then the conditions (i)¨C(iii) hold." See Spaces 7.3.3 and 7.3.4.

Lecture 4: Simple functions

• 4.1 Introduction
• 4.2 Integrals
• 4.3 Estimates

Week 5

The results obtained represent the culmination of our work so far. We will prove three results which are cornerstones of measure theory: The Monotone Convergence Theorem (MCT), Fatou's lemma and the Dominated Convergence Theorem (DCT).

We cover Section 7.5 (Lecture 5) and 7.6 (Lecture 6) in Spaces. Section 7.9 (on complex-valued functions) is not covered completely by the videos.

Lecture 5: Integrals of non-negative measurable functions

• 5.1 Introduction
• 5.2 Basic properties (and bootstrapping)
• 5.3 The Monotone Convergence Theorem
• 5.4 Fatou's Lemma

Remark. We skipped "Comparison with Riemann integration" and will return to this later.

Lecture 6: Integrable functions

• 6.1 Introduction
• 6.2 Basic properties
• 6.3 The Dominated Convergence Theorem
• Self-study: Complex-valued functions. Section 9 up to and including 7.9.1 in Spaces.

Remark. We skipped Proposition 7.6.6 and will probably not return to it later.

Week 6

We cover Section 8.1 and 8.2 in Spaces. Note that since we are reusing lecture recordings from 2021, both the numbering of the lectures and the numbering of the theorems are out of sync. Be advised that we cover less material than usual this week and more material than usual next week.

Lecture 7: Outer measures

• 7.1 Introduction
• 7.2 Basic properties
• 7.3 *-measurable sets

Lecture 8: (X,M,mu*) is a measure space

• 8.1 Introduction
• 8.2 M is a sigma-algebra
• 8.3 (X,M,mu*) is a complete measure space

Week 7

Theorem and lecture numbering remain out of sync in the videos. We finish Spaces by covering Section 8.3 and 8.4.

Lecture 9: Carath¨¦odory's extension theorem

• 9.1 Introduction
• 9.2 Carath¨¦odory's extension theorem for algebras
• 9.3 sigma-finite measure spaces
• 9.4 Carath¨¦odory's extension theorem for semi-algebras (Self-study: pp. 300¨C302 in Spaces.)

Lecture 10: The Lebesgue measure on the real line

• 1.6 Lebesgue measure (repetition from week 3)
• 10.1 Introduction
• 10.2 Constructing the Lebesgue measure
• 10.3 Translation invariance
• 10.4 The Lebesgue measure on intervals
• 10.5 A non-measurable set. (Self-study: pp. 306¨C307 in Spaces.)

Remark. Lecture 10.4 refers to Lecture 5.5, which does not exist. We will cover the same material in Lecture 11.

Week 10

Lecture 15 & 16: Lp spaces

The "real" Lecture 15 and 16 are the physical lectures in week 10, which cover Chapter 2 in ELA. The videos below are only intended as as supplement to the physical lectures. Beware that the numbering of the results in the videos refer to the previous version of ELA. Results in the current version of ELA which are not covered by the videos are: Propositions 2.1.4, 2.1.6, 2.1.7 and 2.1.8. 

• 15.1 Introduction 
• 15.2 L1 is a Banach space
• 15.3 Lp and H?lder's inequality
• 15.4 Minkowski's inequality
• 15.5 Lp is a Banach space
• 15.6 Essentially bounded functions
• 15.7 Linfty is Banach space
• 15.8 Sequence spaces (see also this note from 2020).

Week 13

The physical lectures on 28 March and 29 March, which were intended to cover Section 3.4 and 3.5 in ELA, are replaced by the videos enclosed below. With the exception of the first one, they were all produced in 2021. The theorem numbering is slightly off since what is currently Section 3.4 and 3.5 in ELA used to be Section 4.3 and 4.4.

The lecture notes for the lectures I intended to give can be found in the schedule. I consider the lecture notes to be the canonical presentation of the material, but there are only minor differences between this and what is covered in the videos below. 

Lecture 21: Adjoint operators

• 21.1 The Riesz Representation Theorem
• 21.2 Introduction
• 21.3 Existence and uniqueness
• 21.4 Examples
• 21.5 Basic properties
• 21.6 Operator algebras

Lecture 22: Self-adjoint operators

• 22.1 Introduction
• 22.2 Orthogonal projections are self-adjoint
• 22.3 Numerical range and radius
• 22.4 Positive operators
• 22.5 Carleman's operator
• 22.6 Spectral theorem for hermitian matrices

Week 14

We cover Section 3.6, Section 4.1 and the first part of Section 4.2 in ELA. The first lecture is shorter than usual, while the second is longer. The numbering is again slightly off since the videos are from 2021, but these sections of ELA have not undergone many changes.

Be advised that the first lecture covers slightly less material than usual, while the second covers slightly more.

Lecture 23: Unitary operators

• 23.1 Introduction
• 23.2 Linear isometries
• 23.3 Unitary operators

Lecture 24: Compact operators

• 24.0 Repetition: 
  • Proposition 1.3.2 and the definition of finite rank operators from Lecture 13.
  • Example 3.5.10 from Lecture 22.
  • Exercise 3.20 from week 13.
• 24.1 Introduction
• 24.2 Compact operators are bounded
• 24.3 The subspace of compact operators I
• 24.4 The subspace of compact operators II
• 24.5 The subspace of compact operators on a Hilbert space
• 24.6 Summary (Exercise 4.18 referred to here is Exercise 3.20 mentioned above.)

Week 16

Lecture 25: Hilbert¨CSchmidt operators

In 2021, this material was covered after ELA Chapter 4.3. This means that the weekly exercise discussed at the end of 25.2 below is not currently relevant.

• 25.1 Introduction
• 25.2 Definition
• 25.3 Basic properties
• 25.4 Matrix representations
• 25.5 Integral operators with continuous kernels