• Foundations of Mathematical Analysis, 7.5 hp

    SyllabusThe 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.

    Textbook: Rudin, Principles of Mathematical Analysis, McGraw-Hill, third edition.
    Complemetary material: Ekedahl, Läsanvisningar till Analysens grunder, en, sv.

    Time and Place: August 31–December 11, 2015.  Distance.

    Instructor: Yishao Zhou

    Exam:  It consists of two parts (i) a written part: three sets of homeworks, and (ii) an oral part.  Roughly speaking, to pass the course you are required a satisfactory homework handed in on the assigned due date,  and thereafter a successful oral which will take place after the last lecture.  Delay for hand-ins is allowed but it will reduce your course grade.  The details of grading can be found in grading policy.

    Practical notes:  The course will be running for about 15 weeks.  An oral exam takes place in the week starting December 14.  Each topic (12 in total) below corresponds to a quarter of the full time study week and three weeks for doing homework (PS1, PS2, PS3) which should be turned in on due time.  They are divided in three blocks which will be apparent when you get your problems sets. 

    There is a News Forum which we can use for course information and suggestions from you and a Discussion Forum where you can ask questions, answer questions and discuss problems and ideas you have.

  • 1. Real numbers and countable sets

    We start with recapitulation of different number systems, in paricular, 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)

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

  • 2. Metric spaces and compact sets

    After this week you should be able to do some exercises concerning some basic concepts in topology.  The main topics are metric spaces and compact sets.  There are many definitions which are essential in analysis.  We try to help you to understand them by working on examples, in addition to proving theorems.

    Text: 2.15-2.33

    Exercises: 2:5, 7–9, 12–14.

  • 3. Properties of compact and connected sets

    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: 2.36–2.42, 2.43–2.44 (optional), 2.45–2.47

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

    For you who plan to study (chaotic) dynamical systems I have made an effort to collect some properties of the Cantor set.

  • 4. Numerical sequnces and series

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

    Text:  Ch.3

    Exercises:  Ex. 3:2, 4–5, 7–9, 12, 13, 16, 20, 23–24

  • 5. Continuity

    It is similar to study continuity of a function defined on a metric space as that on the real numbers, namely by the notion of limits.  Thus we need a definition of limits on metric spaces, and the properties.  We remind you of the nice properties of a continuous function defined on an interval (in our earlier terminology, a closed interval).  On metric spaces it is natural to get these properties on a compact set.  Finally we deal with the notion of discontniuity. 

    Text:  Ch.4

    Exercises:  4:2–3, 7–8, 10, 14, 18–19

  • 6. Differentiation and the Riemann-Stieltjes integral

    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 (quite a lot).  Basically you'll do the same thing as you have done in the previous analysis/calculus courses.  I only set up some questions on differentiation. 

    Text: Ch. 5, Ch. 6,  6.23–6.27 (optional)
    Exercises: 5:1–2, 6, 11, 17, 19; 6:1–2, 4, 7–9.

    Note: Although the 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 we have will deal with such things.

  • 7. Uniform convergence and continuity

    The main problem to deal with is the limit processes, in particular, the interchange of limit processes.  In order to reach some positive results a notion on uniform convergence is introduced.  We investigate

    • uniform convergence and continuity
    • uniform convergence and integration
    • uniform convergence and differentiation

    This time we do the first part.

    Text:  7.1–7.15

    Exercises:  7:1, 3, 6–7, 9, 11.

  • 8. Interchange of limit processes.

    Let us now continue our investigation on interchange of limit processes.  We'll study uniform convergence and integration, uniform convergence and differentiation, and finally we prove the Weierstrass theorem: A countinuous 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

    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.

  • 9. Functions of several variables, I

    We shall deal with the differentiation of functions defined on 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

    Exercises: 9:6, 8, 9, 14–15

  • 10. 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

  • 11. Functions of several variables, III

    By now we know the general principle that the local behavior of a continuous differentiable mapping F near a point x is similar to that of the linear transformation F'(x).  We'll illustrate this principle by the rank theorem.  As for functions of one variable we can define derivatives of higher order and differentiation of integrals.

    Text:  9.30–9.37, 9.38–9.43

    Exercises: 9:27–29

  • 12. Function spaces

    A text of Introduction on Functional analysis and its application

    Reading. Chapters 1, 2 and 4.

    Exercises: 1: 1-5; 2: 1-3; 4: Even numbers.

    Note: You will get extra credits if you hand in solutions to some of these excerises along with your last set of homework.