# TMA4225 Foundations of analysis: Fall term 2020

## Lecturer

## Lectures

Mondays, 08:15-10:00, S1

Tuesdays, 08:15-10:00, Galleri 4 (might be changed)

Attendance list:

Tuesdays IMPORTANT NEWS ABOUT THE EXAMINATION: there will be a written home exam on Thursday December 10 This will be followed by an adjusting oral exam on Thursday December 10 14:-16:00 Friday December 11 9:00-12:00 and 13:00- 16:00 and finally on Monday December 14 9:00-12:00 and 13:00-16:00. You will have to sing up for one of these times by sending me an email: berit.stensones@ntnu.no Information about the rooms will come later ALERT WE MAY BE FORCED TO GO DIGITAL PLEASE COME TO CLASS TOMORROW DECEMBER 8 SO WE CAN DISCUSS HOW BEST TO DO THIS. LAST NEWS: The oral will have to be done on ZOOM. LST CHANCE TO SIGN UP FOR THE ORAL: Friday November 20

## Exercise sessions

Fridays, 14:15-15:00, Galleri 4 Exercises for week 35 from Axslers book 2A: 1,6,10,11 2B: 6,9,11 Exercises for week 36 : 2B: 12,13,15,23,28 2C:2,4 Exercises for week 37 : 2C:10,11 2D:4,6,7,8,10,19 Exercises for week 38 : 2E: 3,4,5,9,10,15 3A:2,3,5,7 Exercises for week 39: 3A:9,11,15,19 3B:2,3,5,12,13 Exercises for week 40: 4A:1,2,4,8,14 4B:1,3,5,9,10 Exercises for week 41: 5A:1,2,4,8 5B: 1,2,3,4 Exercises for week 42: We will work on the problems from 5B Exercises for week 43: we will work on 5C:4,5,8,11,12,14 Exercises for week 44: We will work on 7A:1,4,6,8,12,13 Exercises for week 45 : 7B : 3,4,5,12,16,17 AFTER THE REVIEW WE WILL COVER THE FOLLOWING OLD EXAMS: Look for them on last years webpage. December 2004, December 2008, December 2011,December 2013. SUGGESTED EXAM PROBLEMS: December 2003 : 2,4,5 Summer 2006 :1,3,4,5 December 2007 :1,2,3,5 December 2010 : 1,3,4 December 2014 : 1,2,3

## Textbook

- Terence Tao:
*An introduction to measure theory*. A draft copy is freely available on the author's webpage, click here for the pdf file.

- Sheldon Axler:
*Measure, Integration and Real Analysis*. The electronic version is available without cost, click

http://measure.axler.net here to find the pdf file.

I have come to the conclusion that the Tao book has many nice ideas, but it is not well suited to study from for a student new to this topic. With help from fellow mathematicians I have found a very well structured textbook and choose to switch to that. Hence from now on there will be the Axler book which will be the main text book. A list of chapters that will be covered will be filled in within a day or two

## Syllabus

Chapters 2,3,4 and 5 from Sheldon Axlers book

In week 35 we covered 2A and parts of 2B In week 36 we will continue covering 2B and do parts of 2C In week 37 we will cover material from 2 D and 2E including Egorov's theorem In week 38 we will finish off the material from 2E and also cover section 3A In week 39 I will cover section 3B and parts of section 4A We will also discuss examples that are not in the text In week 40 we will talk about the material fro section 4A and 4B In week 41 we will cover 5A, 5B and parts of 5C I will change the proofs and definitions a bit from the book We have gone a bit slower then anticipated so in week 42 we will continue with the material from Chapter 5. Future plans for the course: After reviewing some material in Banach space we will study L^p spaces. In week 43 we will finish the material in section 5C and also start a review on Banach Spaces In week 44 we will talk about the material in section 7A and some from 7B In week 45 we will finish 7 b.We shall also start the review MATERIAL THAT WILL BE COVERED ON THE EXAM: Chapter 2 all, Chapter 3 all, Chapter 4 except the part that starts with 4.20, Chapter 5 all, Chapter 7 all