Topic outline

  • Foundations of Mathematical Analysis, 7.5 hp

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    Syllabus:  The course deals with real numbers, theorems on continuous functions on compact sets, derivation and integration in Rn, series of functions, uniform convergence, implicit functions. The course aims at giving a deeper understanding of the foundations of real analysis.

    Instructor: Sofia Tirabassi tirabassi@math.su.se

    Teaching Assistant: 

    • Oliver Lindström V25-V27
    • Jacob Kuhlin V28-V30
    • Kilian Liebe, V31-V34




    You can contact us through the Forum of the corurse-page, or by sending a message through the course-page. 

    Textbook:  [R] Rudin, Principles of Mathematical Analysis, McGraw-Hill, third edition. A Solutions manual to the exercises in the coursebook can be found here.
    The book Real Mathematical analysis  by Charles Pugh, cover more or less the same material (but not the Stiljes Integral) and can be used for the theory instead of Rudin. The suggested eercise are from Rudin's book, but it is a god exercise to try to solve also Pugh exercises.

    You may also find it very useful to have, by your side, the following book (you may download an electronic version of it from the SU library) for the first four weeks:

    Complementary material:

    Reading material:

    • A hand-in with some (important) tips on how to read the course-book, en
    • Ekedahl, Läsanvisningar till Analysens grunder, en, sv
    • T. Tao, Analysis 1

    Important Dates
    The course starts on week 24, and will be finished by week 34.

    The intended dates for releasing each homework, the respective hand-in dates jointly with the material that could be covered in parenthesis are the following ones:

    Homework 1 Due June 29th
    Homework 2 Due July 14th 
    Homework 3 Due July 28th
    Date of Written Exam: Thursday August 8 
    (Note: The times and dates in the course refer always to the corresponding ones in the Swedish time zone)

    Recall to sign-up for the written exam (at the SU campus) at least 14 calendar days before the date. Read more on how to sign-up here.
    NB: If you don't have the possibility of taking the exam in Stockholm read the following page (In Swedish). Make arrangements with good time.

    Date of the Resit-Exam: Wednesday 23 October

    Examination

    The details of grading and examination can be found in the document grading criteria,  but we highlight below some important parts.

    The grade of the course (G) will be over 30 points (0 ≤ G ≤ 30).

    All students should take a written exam, which will consist of up to 5 problems. The score obtained
    in the written exam (W) will give a maximum of 24 points (0 ≤ W ≤ 24).

    Those students that obtain a score greater than 21 in the written exam (21 ≤ W ≤ 24) have the
    right to take an oral exam. The score of the oral exam (O) will give a maximum of 6 points
    (0 ≤ O ≤ 6).

    During the course three homework will be released, so that each student can obtain bonus points
    (B). Each homework will be graded over 1 point, so that a maximum of 3 bonus points can be
    obtained (0 ≤ B ≤ 3).

    • If a student obtains a score smaller than 21 in the written exam (0 ≤ W < 21), the grade will be given by G = W + B.
    • If a student obtains a score greater than 21 in the written exam (21 ≤ W ≤ 24), the grade will be given by G = min (W + B, 24) + O.

    Monitoring
    Any question regarding the course material, doubts or enquiries shall be written preferably in the general Discussion forum, and those of a more private nature, by using the message system in the course-page. Email messages concerning the course content are strongly discouraged, and would be preferred them to be posted in the Discussion forums for public discussion. Those will be regularly monitored by the teachers, and questions posted there will be answered accordingly.

    Homework
    Note that every handed-in solution should consist of a single pdf file, and must be typed in LATEX (a good and free of charge tool for collaborative typing in latex is www.overleaf.com). The handed-in homework, should be submitted only by using the submission system that will be found in the course-page in due time. Late submissions will not be allowed.

    Written exam
    The written exam will take place on Thursday August 8th

    Oral exam
    The oral exam will take place onthe 16th of August on Albano premises. (Eventually also on the 17/8).

    Those qualifying for taking the oral exam (see Grading policy), will have the chance to book a time for that. A Doodle will be published to that effect on Friday the 12th of August.  The form should be filled no later than Sunday the 14th of August (23:59 at the latest).

    On Monday the 15th of August a list of booked time and examiners will be published in the course-page.


    Meetings:

    I will try to be available via zoom once a week at the beginning of the week to answer questions/ lecture you about the course material. The time  will change. I will try to record these lecture an put the vidoe online. The time for the meeting will be in the tab of the corresponding week.

    Meeting ID 68536715350


  • 0. Introduction to the course

    Welcome to the course!

    During this week previous to the start of the course, we recommend you to go through the course-page and documents. 

    Please, familiarise yourself with the course-page, and create/update your personal profile uploading, if possible, a picture of yourself. Read all the course instructions, and specially make sure that you have read carefully all the documentation regarding grading policy and examination, as well as the intended weekly study plan.

    If you haven't done  it yet, please formalise your registration for the course.

    One of the main goals of the course is for the student to learn to read and understand mathematics. So, this is mainly a distance reading course.   To be able to read mathematics one must

    • learn to understand the formal and precise mathematical language,
    • create one’s own language (vital) or images and
    • learn to translate between the common formal language and one’s private.
    Reding mathematics is not like reading novels, as it requires the active implication of the reader. To help you in the process, we have prepared a hand-in with some (important) tips on how to read the course-book.
    Please have a look to this document.


  • 1. Real numbers and countable sets

    Zoom meeting:  June 11th 9:15-11


    Lecture video 1

    Lecture video 2


    We start with recapitulation of different number systems, in particular, the real and complex numbers.  Then we study the topics such as ordered sets and fields, commonly in this course the real field and complex field.  A difficult notion to understand, and basic in mathematical analysis is supremum/infimum.   This notion is not completely new to us.   If you want to understand the real number system further, it is almost a must to know the Dedekind's construction. 

    When we are ready we turn to the basics in topology.  The first attention is on  countable sets and uncountable sets. 

    Text: Chapter 1, 2.1–2.14, (Appendix to chapter 1 optional)

    Chapters: 4,5,6,7, 8 in [Ra]. 

    Suggested exercises:  1:1–3, 5–6, 8–9, 12–15; 2:2–4.

    Extra activities: Watch the video on the Infinite Hotel Paradox.

    The Infinite Hotel, a thought experiment created by German mathematician David Hilbert, is a hotel with a countable number of rooms.

  • 2. Metric spaces and compact sets

    Zoom Tuesday 8:30-1030 (lecture video follow the link)

    The main topics are metric spaces and compact sets.  There are many definitions which are essential in mathematics, and in particular those for open, closed and compact sets. 

    Text: 2.15-2.33

    Suggested Exercises: 2:5, 7–9, 12–14.

    [Ra] Chapters 9,10 and 11 (until Theorem 11.17)

    The first homework is released.


  • 3. Properties of compact and connected sets

    Zoom Monday 9-11

    Lectue recording

    We continue studying compact sets and their properties this week.  Among others we shall prove that

    • A compact set is always bounded
    • A compact set is closed
    • Weierstrass theorem

    Then we turn to the notion of connected sets.

    Text: 2.34–2.42, (Optional: 2.43–2.44), 2.45–2.47

    [Ra] Chapters: 11,12,13

    Exercises: 2:15–16, 19, 21–22, 29.

    For you who plan to study (chaotic) dynamical systems, you can find here some properties of the Cantor set.


  • 4. Numerical sequences and Series

    Zoom Tuesday July 2 13-15Lecture recording

    Zoom Tuesday July 2 13-15Lecture recording Sorry, I forgot to turn on the recording at the beginning, so a little bit is lost



    Now we turn to sequences and series of complex numbers in a metric space.  This week we shall study convergence of sequences, subsequences and sequences with the Cauchy property. We should also focus our attention on some special type of sequences and series.  We also introduce the notions of upper limits and lower limits.  We'll pay special attention to the root and ratio tests, summation by parts, and rearrangement of series.

    Text:  Chapter 3 (Optional: 3.52-3.55)

    [Ra]. Chapters 14,15,16,17,18,19

    Suggested exercises:  Ex. 3:2, 4–5, 20, 21, 22, 23, 24

    The second homework is released..

  • 5. Continuity

    Meeting: Monday July 8th kl 8-10

    Link for the recording

    It is similar to study the continuity of a function defined on general metric spaces as that on the real line by using the notion of limit.  Thus we need to define the concept of limit on metric spaces and study its properties.  We shall see the nice properties satisfied by continuous functions that are defined on a compact set (in our earlier terminology, a closed interval). Finally, we deal with the notion of discontinuity.

    Text:  Ch.4

    Exercises:  4:2–3, 7–8, 10, 14, 18–20


     

  • 6. Differentiation and the Riemann-Stieltjes integral

    Meeting: Tuesday  July 16th kl 8:15

    Video Recording


    The topics of differentiation and integration are minor and less heavy than the other parts of this course. So we'll quickly go through them.  Basically you'll do the same thing as you have done in the previous analysis/calculus courses. The newest topic you will see in this chapter is the concept of integration of Riemman-Stieljes, which is more general than the Riemann integration that you have seen in previous courses.

    Text: Ch. 5, Ch. 6. (Optional: 6.23–6.27)

    Exercises: 5:1–2, 6, 11, 17, 19; 6:1–2, 4, 7–9.

    The third homework is released.


    Note: Although the integration of vector-valued functions are not required in this course, it is highly recommended that you read through the material provided here because in many mathematical subjects one has to deal with such tools. Note that sections 6.23–6.27 are optional.

  • 7. Uniform convergence and continuity

    Meeting: Tuesday July 23th kl 8:15

    Lecture video

    We would like to deal with the problem of interchanging limit processes.  However, before obtaining some positive results, one has to introduce the notion of uniform convergence. This chapter is devoted to the study of uniform convergence and continuity.

    Text:  7.1–7.15

    Exercises:  7:1, 3, 6–7, 9, 11


  • 7 II Interchange of limit processes.

    This chapter is devoted to the study of interchanging limit processes.  More precisely, we study uniform convergence and differentiation as well as uniform convergence and integration. Finally, we shall prove the Stone-Weierstrass theorem: A continuous function on [a,b] can be approximated by a sequence of polynomials uniformly. 

    Text:  7.16–7.18, 7.26

    Exercises: 7:12, 20, 23

    IMPORTANT: Recall to register for the written exam (at the SU campus) at least 7 calendar days before.

    Note that Rudin chooses to deal with these problems when limit processes are interchanged for complex-values functions in order to provide the most important aspects.  Many theorems can be extend to vector-valued functions and further to mappings into general metric spaces.  These will be topics in courses such as set point topology.  You can try to do some if you are interested in the topics.

  • 8. Functions of several variables, I

    We shall deal with the differentiation of functions defined on the Euclidean space {\mathbb R}^n.  This is, roughly speaking, defined by a suitable linear transformation.  Here we need some matrix computation and properties of linear transformations.

    Text:  9.1–9.21 (9.22-9.29 Without proofs)

    Exercises: 9:6, 8, 9, 14–15


    IMPORTANT: Recall to register for the written exam (at the SU campus) at least 10 calendar days before.

    For those interested, I include some videos where I go through the proof of the Implicit and Inverse Function theorems. 

    In the coursebook, Rudin proofs the Inverse Function Theorem, and deduces from it the Implicit function Theorem. In the videos below, I present those in an alternative way, proving first the Implicit Function Theorem, and deducing from it the Inverse Function Theorem. 

  • 8 ii. Functions of several variables, II

    The purpose of this lecture is to prove the three very important theorems,  the contraction principle, the inverse function theorem and the implicit function theorem.  They are very useful in many mathematical disciplines.  Take time to think about them and try to understand them by working on concrete problems.

    Text:  9.22–9.29

    Exercises:  9:17, 20, 23–24

  • Oral Exams