TMA4225 Analysens grunnlag – Foundations of Analysis
Lecturers
- Office: room 1156, Sentralbygg 2
and
- Office: room 1129, Sentralbygg 2
Textbook
- Sheldon Axler: Measure, Integration & Real Analysis, Springer 2020. Click here for the pdf file (from NTNU accounts). A paperback version can be purchased from Springer for € 39.99. A free pdf is available from the author's homepage.
Reference group
Niels Christiansen nielsic [at] stud [dot] ntnu [dot] no; Henning Hegstad henniheg [at] ntnu [dot] no
Lectures
Wednesday 8:15-10:00, GL-GE G21
Thursday 14:15-16:00, GL-GE G21
Exercises (Øvinger) Wednesday 14:15-15:00, GL-OP B3
The lectures started in week 34
The exercises started in week 35
The lectures in weeks 34-35 were given by
Magnus B Landstad
* Office: room 1129, Sentralbygg 2
* Email: magnus [dot] landstad [at] ntnu [dot] no
I suggest that you take a look at what you know about Riemann integration. Look in your calculus books or at the beginning of
https://measure.axler.net/SupplementMIRA.pdf
Also read the Preface for Students in the book and I will read the the Preface for Instructors.
| Week | Topic | Notes |
|---|---|---|
| 34 | Suppment Section A: Parts without proofs Section C: Parts, only for \(\mathbb{R}\) Section D: Most of this is supposed to be known for R. (0.59 may be taken later). Section E: Some of this is should be known for R. (0.77–0.79 may be taken later). Textbook Ch 1: Mostly without proofs Ch 2: 2.1, 2.2, 2.10, and 2.12 (Heine–Borel) | |
| 35 | Ch 2: 2.1–2.25, 2.27 | |
| 36 | Ch 2 B, C: 2.26–2.66, 2.68 | Skipped (covered in week 37) 2.31–2.53. 2.66 and 2.68 could bear repetition – I was a bit rushed at the end. My proofs (especially for 2.62) were a bit different from those in the book. This little note has the details. |
| 37 | Wed: Covered p. 29-37 (inverse images and measurable functions.) Thu: Cantor set and Cantor function, and then Sec. 2E, ending with 2.85 (Egoroff's theorem). | Egoroff's theorem is summarized in Littlewood's third principle: Pointwise convergence is almost uniform convergence. |
| 38 | Wed: 2.88–2.93 (Luzin's theorem). Thu: 2.94–2.95 (Lebesgue measurable functions), then on to Ch 3A (integration) ending with 3.11 (Monotone Convergence Theorem, MCT) | I have my own take on the initial part of Ch 3. Here is a short note explaining it. |
| 39 | Wed: Backtracked a little, starting with 3.9 and repeated the basic idea of the MCT. Continued up to 3.25 (also included 3.27) Thu: The big limit theorems, starting with 3.26 (BCT), also conditions for Riemann integrability, ending with 3.39. Plus 3.40 and a a preview of what's up next. | |
| 40 | Wed: Finished Ch 3, then Ch 4.A to 4.4 (Vitali covering lemma) Thu: Finished Ch 4. | My version of the proof of the Lebesgue differentiation theorem: It is essentially the same proof as in the book, but organized to have fewer moving parts. |
| 41 | Wed: Ch. 5, pp. 116-123. Thu: pp. 123-130. | |
| 42 | Wed (Harald): pp. 131–139, note ➔ Thu (Helge): Ch. 5: pp. 140-143. (only proved the equality of mixed derivatives.) Ch. 7: pp. 193-197. | Here is a note on uniqueness that I used. |
| 43 | Wed: Ch. 7, pp. 198-204. We stopped in the middle of the proof of Thm. 7.20. Thu: The regular lecture was replaced by the 2024 Lars Onsager Lecture, presented by Sir Martin Hairer, who received the Fields Medal in 2014. Lecture title: Taming Infinities |
|
| 44 | Wed: Ch. 7, pp. 204-205. Ch. 12, pp. 380-384. Thu: Ch. 12, pp. 385-389. | |
| 45 | Wed: Ch. 12, pp. 390-393. Thu: Ch. 12, pp. 393-397. | |
| 46 | Wed: Ch. 9, pp. 268–272. Thu: Ch. 9, pp. 273–276. | |
| 47 | Wed: Ch. 9, pp. 276-277 and 256-259. Thu: Exam TMA4225 from 2023, except Problem 2 and 5c. | Exam 2023 ("I_A" means the characteristic function or indicator function of the set "A") |
Curriculum
Curriculum (all from Axler): Chapters 1-5, 7, 9 (pp. 256-259, 268-277), 12.
Exam
Written exam 19 December. Time: 4 hours. Venue: TBD
Permitted aids: (Support material code D) No printed or hand-written support material is allowed. A specific basic calculator is allowed.