Week 19
In the first lecture, we briefly discuss some applications of Fredholm theory. This is the final lecture with new material. In the second lecture, we review the second part of the course (Lecture 17 to Lecture 32).
Lecture 33: Applications of Fredholm theory
- 33.1 Introduction
- 33.2 Integral equations
- 33.3 Ordinary differential equations (the video is from Spring 2020, you can start at the six minutes mark).
Lecture 34: Hilbert spaces review
- 34.1 Hilbert spaces review: some key concepts and results
Week 18
In the first lecture, we wrap up the last part of Section 5.2 in ELA. We then begin with Section 5.4.
Lecture 31: Hilbert–Schmidt operators
- 31.1 Introduction
- 31.2 Definition
- 31.3 Basic properties
- 31.4 Matrix representations of operators
- 31.5 Integral operators with continuous kernels
Lecture 32: The Fredholm alternative
- 32.1 Introduction
- 32.2 What is the Fredholm alternative?
- 32.3 mu is not an eigenvalue
- 32.4 mu is an eigenvalue
Week 17
We postpone the second part of Section 5.2 in ELA to next week and spend this entire week Section 5.3. Our focus is the spectral theorem for compact self-adjoint operators on a Hilbert space, which is the most important result of the last part of the course.
Lecture 29 & 30: The spectral theorem for compact self-adjoint operators
- 29.1 Introduction
- 29.2 Three lemmas
- 29.3 Proof of (a), (b) and (c)
- 29.4 Proof of (d) and (e)
- 29.5 Proof of (f)
- 29.6 Multiplicity of eigenvalues
- 29.7 Self-study: Diagonalizable operators (pp. 95–96 in ELA).
Week 16
We first wrap up Chapter 4 in ELA, then cover Section 5.1 and the first part of Section 5.2. (Note that Lecture 27 covers less ground than usual, while Lecture 28 covers more.)
Lecture 27: Unitary operators
- 27.1 Introduction
- 27.2 Linear isometries
- 27.3 Unitary operators
Remark. We skipped Theorem 4.5.9 in ELA and will return to it if needed.
Lecture 28: Compact operators
- 19.2 Summary of last time+compact operators (repetition from week 11)
- 26.6 The spectral theorem for Hermitian matrices (repetition from week 15)
- 28.1 Introduction
- 28.2 Compact operators are bounded
- 28.3 The subspace of compact operators I
- 28.4 The subspace of compact operators II
- 28.5 The subspace of compact operators on a Hilbert space
- 28.6 Summary
Week 15
We cover Section 4.3 and Section 4.4 in ELA. The emphasis is on the adjoints of bounded operators and self-adjoint operators.
Lecture 25: Adjoint operators
- 25.1 Introduction
- 25.2 Existence and uniqueness
- 25.3 Examples
- 25.4 Basic properties
- 25.5 Operator algebras
Lecture 26: Self-adjoint operators
- 26.1 Introduction
- 26.2 Orthogonal projections are self-adjoint
- 26.3 Numerical range and radius
- 26.4 Positive operators
- 26.5 Carleman's operator
- 26.6 The spectral theorem for Hermitian matrices
Week 14
This week we go through the Riesz representation theorem. This is an important theorem with many applications that you will see in later courses.
Lecture 24: The Riesz representation theorem
- 24.1 Introduction
- 24.2 The Riesz Representation Theorem
- 24.3 Examples and non-Examples of Riesz
Week 13
This week we go through chapter 4.2 in ELA. Unlike last time I now follow ELA more or less exactly. Since there is only one lecture this week I won't give exercises this week, but postpone them till next week.
Lecture 23: Orthonormal bases
- 23.1 Introduction
- 23.2 Examples of orthonormal bases
- 23.3 The Gram Schmidt Process
- 23.4 Two lemmas (The statement of the second lemma should be that <y, u_k>=c_k for all k, NOT that <y, u_k>=0).
- 23.5 Characterizations of when an orthonormal set is an orthonormal basis
- 23.6 Illustration of Theorem
Week 12
In this week's lecture we go through 4.1 in ELA. Note that I do it slightly differently than ELA in that I divide Theorem 4.1.2 into two parts.
Lecture 21 & 22: Orthogonal projections and Hilbert space decompositions.
- 21.1 Introduction
- 21.2 Closed and convex sets in Hilbert spaces
- 21.3 Main Result 1
- 21.4 Main Result 2
- 21.5 An example and some Corollaries
- 21.6 L^2 as a Hilbert space
Week 11
This week we finish chapter 3 in ELA. The first lecture explains complemented subspaces and bounded projections for normed spaces and how there is a one-to-one correspondance between them for Banach spaces (this fact is not completely proven in ELA, but I include it). In the second lecture we discuss decomposable operators and extension of linear operators by continuity. The latter is especially important and I encourage you to understand the proof fully.
Lecture 19: Complemented subspaces and associated projections
- 19.1: Introduction
- 19.2: Summary of last time+compact operators
- 19.3: Complemented subspaces
- 19.4: Bounded Projections 1
- 19.5: Bounded Projections 2 (It should really have been a part of the theorem statement that P and Q are defined by P(m+n)=m and Q(m+n)=n for m\in M and n\in N. This is the "obvious" way to define projections when X is the direct sum of M and N).
Lecture 20: Decomposable operators & extension by continuity and density
- 20.1: Introduction
- 20.2: Decomposable operators
- 20.3: Extension by continuity
Week 10
We begin on the second part of the course which primarily deals with normed spaces and Hilbert spaces and bounded linear operators defined thereon. The first lecture is a revision of stuff you've seen before and is foundational for the rest of the course. The second lecture explains how things are "nice" for finite dimensional normed space.
Lecture 17: Revision of normed and -inner-product spaces and characterizations of boundedness
- 17.1 Normed spaces
- 17.2 Inner-product spaces
- 17.3 Bounded Operators
Lecture 18: Finite dimensional normed spaces
- 18.1: Norms on finite dimensional spaces are bounded (NB: In the proof of the first theorem I want M and K to be the minimum and maximum of the norm on the unit sphere, not the unit ball.)
- 18.2: Further results
- 18.3: Final result and an example
Week 9
We wrap up the measure theory part of the course. The first lecture contains a selection of topics which are not adequately covered in Spaces or ELA. The second lecture is a short review of some key concepts and results.
- Office hour for the mandatory assignment: Monday 09:15–11:00 (see Zoom).
Lecture 15: Measure theory addendum
- 15.1 Introduction
- 15.2 The Riemann–Lebesgue lemma
- 15.3 Bounded linear functionals on Lp
- 15.4 Absolutely continuous measures
- 15.5 The Cantor set
Lecture 16: Measure theory review
- 16.1 Measure theory review: Some key concepts and results
Week 8
In the first lecture, we will finally construct the Lebesgue measure on the real line (Section 8.4 in Spaces). The second lecture will be based in part on Section 8.5 in Spaces.
- Office hour: Monday 14:00–15:00 (see Zoom).
- Office hour for the mandatory assignment: Friday 09:15–10:00 (see Zoom).
Lecture 13: The Lebesgue measure on the real line
- 1.5 Lebesgue measure (repetition from week 2).
- 13.1 Introduction
- 13.2 Constructing the Lebesgue measure
- 13.3 Translation invariance
- 13.4 The Lebesgue measure on intervals
- 13.5 A non-measurable set. Self-study: pp. 306–307 in Spaces.
Remark. If you are not planning to continue with MAT4410 – Advanced Linear Analysis in the fall, then I strongly recommend you to read Sections 8.7 and 8.8 in Spaces. (Feel free to skip the proofs.)
Lecture 14: Approximation results
- 14.1 Introduction (and Littlewood's three principles)
- 14.2 Littlewood's first principle
- 14.3 Dense subsets in Lp
- 14.4 Littlewood's second principle
Week 7
We wrap up Section 8.2 from Spaces in the first lecture and go through Section 8.3 in the second. (Note that Lecture 11 covers less ground than usual, while Lecture 12 covers more.)
- Office hour: Monday 14:00–15:00 (see Zoom).
Lecture 11: (X,M,mu*) is a measure space
- 11.1 Introduction
- 11.2 M is a sigma-algebra
- 11.3 (X,M,mu*) is a complete measure space
Lecture 12: Carathéodory's extension theorem
- 12.1 Introduction
- 12.2 Carathéodory's extension theorem for algebras
- 12.3 sigma-finite measure spaces
- 12.4 Carathéodory's extension theorem for semi-algebras (Self-study: pp. 300–302 in Spaces.)
Week 6
In the first lecture, we wrap up our study of abstract measure spaces by looking at different notions of convergence. The results presented here are mostly from Section 7.8 in Spaces and some leftovers from Section 2.2 in Elements of Linear Analysis. Our next major goal is to construct the Lebesgue measure on the real line, and the second lecture will cover Sections 8.1 and the first part of 8.2 from Spaces.
- Office hour: Monday 14:00–15:00 (see Zoom).
Lecture 9: Convergence, convergence, convergence and convergence
- 9.1 Introduction
- 9.2 What implies convergence in measure?
- 9.3 What does convergence in measure imply?
- 9.4 Egorov's theorem (no sound for the proof part of the video)
Lecture 10: Outer measures
- 10.1 Introduction
- 10.2 Basic properties
- 10.3 *-measurable sets
Week 5
Section 2 in Elements of Linear Analysis and Section 7.7 (and the second half of 7.9) in Spaces cover the same material. The videos below are mostly based on Elements of Linear Analysis, although our exposition will be slightly different.
Lecture 7 & 8: Spaces of integrable functions
- 7.0 The story so far & what's next?
- Self-study: Section 1.1 in Elements of Linear Analysis. These two pages contain some results about normed vector spaces, which I assume have been covered in a previous course. If you need more information, consult Sections 5.1 and 5.2 in Spaces.
- 7.1 Introduction (and seminorms)
- 7.2 L1 is a Banach space
- 7.3 Lp and Hölder's inequality
- 7.4 Minkowski's inequality
- 7.5 Lp is a Banach space (and a Hilbert space if p=2)
- 7.6 Essentially bounded functions
- 7.7 Linfty is Banach space
- 7.8 Sequence spaces (see also this note from last year).
Week 4
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. Parts of Section 7.5 (on the Riemann integral) and Section 7.9 (on complex-valued functions) is not covered completely by the videos.
Lecture 5: Integrals of non-negative measurable function
- 5.1 Introduction
- 5.2 Basic properties (and bootstrapping)
- 5.3 The Monotone Convergence Theorem
- 5.4 Fatou's Lemma
- 5.5 The Riemann integral (Self-study: pp. 267–268 in Spaces.)
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 7.9.2 in Spaces.
Remark. We skipped Proposition 7.6.6 in Spaces and will return to it later if we need it.
Week 3
We continue with digital teaching and we will 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)–(iii) hold." See Spaces 7.3.3 and 7.3.4.
Lecture 4: Simple functions
- 4.1 Introduction
- 4.2 Integrals
- 4.3 Estimates
Remark. The material from Spaces 7.4.2–7.4.4 is organized differently in the videos.
Week 2
The government has decided that all teaching will be digital in week 2. We will cover Section 7.1 and 7.2 in Spaces. The videos below cover the material much quicker than a typical lecture. I encourage you to watch them with pen and paper at available and to pause whenever necessary.
Lecture 1 & 2: Measure spaces
- 1.0 Introduction
- 1.1 Sigma algebras
- 1.2 Measure spaces
- 1.3 Null sets
- 1.4 Complete measure spaces
- 1.5 Lebesgue measure