Lecturer: Associate Professor Jerry Koliha

Address: Room 164 in the Richard Berry Building

e-mail: j.koliha@ms.unimelb.edu.au

I welcome you to what I hope will be an enjoyable and instructive semester.

It is important to familiarize yourself with the contents of the prescribed Lecture Notes. The Appendices contain background material for the course. Please read Appendices A and E in the first weeks of the semester. You are welcome to discuss any problems with me outside lectures (in the scheduled consultation hours or at other times by appointment). The first part of the Notes relates to metric spaces; the section on completion of metric spaces gives an alternative approach to the one presented in 620-311 and will be a crucial tool for the construction of the Lebesgue integral. It is expected that students will take advantage of the MacTutor History of Mathematics website containing biographies of mathematicians and a history of mathematical disciplines.

Class Representative: Yi Huang

Visit your lecturer's home page

620-312 Contents

News

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• Assignment 4 is due on 18 September 2007.
  
  
• An amusing historical note spanning 160 years and featuring the infuriating Joseph Fourier and the pointwise convergence of the classical Fourier series: fourier
  
  
• Following a request, this file discussing techniques for finding primitives (indefinite integrals) was posted on the web: primitives
  
  
• Integration: Problem Sheet 1  Problem Sheet 2 Problem Sheet 3 Problem Sheet 4
  
  
• Functional Analysis: Problem Sheet 5 Problem Sheet 6 Problem Sheet 7 Problem Sheet 8
  
  
• Solutions to two past exam papers: exam2003 exam2004
  
  
  
  
  
  
  Important (and interesting) links:
  
  
• The ARC Centre of Excellence for Complex Systems MASCOS - for possibilities in Maths and Stats
  
  
• The Australian Mathematical Institute AMSI - for possibilities in Maths and Stats
  
  
• The Australian Mathematical Society AustMS
  
  
• Biographies of mathematicians (and more): See the University of St. Andrews, Scotland, MacTutor History of Mathematics website
  
  Recommended biographies:
  
  Integration: Bernhard Riemann, Emile Borel, Camille Jordan, Henri Lebesgue, Constantin Caratheodory, Giuseppe Vitali, Hans Hahn, Guido Fubini, Johann Radon, Otton Nikodym, Thomas Stieltjes
  
  Functional Analysis: Ivar Fredholm, Vito Volterra, Frigyes Riesz, David Hilbert, Erhardt Schmidt, Stefan Banach, Hugo Steinhaus, Rene-Louis Baire, Juliusz Schauder, and a host of other mathematicians
  
  
• Hubble space telescope pictures   
  
  
• Mathematical Reviews on line (American Mathematical Society)
  
  
• Want to make million bucks (US, of course)? Click on this link to find how
  
  

Lectures

Contents

If you miss a lecture, you may download the following black and white scans of the lecture transparencies. The scans are in portable document format (PDF) and you will need Adobe's Acrobat Reader or plugin for your web-browser.  The files are protected by a password given in the lectures.

Reading these Lecture Notes is NOT a substitute for attendance. If you want to do well in this subject, you have to attend Lectures and Practice Classes.

Download: Adobe Acrobat Reader

Assessment in 620-312

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The assessment in this subject will be based on four assignments due during the Semester and a three hour examination at the end of Semester. Your mark in 620-312 will be the maximum of

Lecture Times

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Your lecture times (L=lecture, P=Practice Class)

Monday L	4.15-5.15 pm	Room 213, R. Berry Building
Tuesday L	1.15-2.15 pm	Room 213, R. Berry Building
Friday L	2.15-3.15 pm	Room 213, R. Berry Building
Thursday P	10.00-11.00 am	Room 213, R. Berry Building

The consultation hours are for individual consultations. They will start in the second week of Semester. The students can come at other times, subject to the availability of the lecturer.

Thursday 11-12 pm	Room 164 R. Berry Build.
Friday 11-12 pm	Room 164 R. Berry Build.

There will be 4 assignments that will contribute up to 20% towards your final mark in this subject. Assignments will be handed out a week before the due date. Copies will be available through this Web site. Click on Assignment X to download the text of the assignment, and click on the due date to download a solution (understandably, solutions are available only after the due date).

Students who are unable to submit an assignment on time and qualify for special consideration, should contact A/Prof Koliha as soon as possible.

These assignments must be your own work. While students are encouraged to discuss their coursework and problems with one another, assignments must be written up independently. According to the University Council ruling, each student is required to hand in a signed statement regarding plagiarism for each subject. Download the file here: plag.pdf

Assignments	Due date
Assignment 1	16 August 2007
Assignment 2	6 September 2007
Assignment 3	13 September 2007
Assignment 4	18 October 2007

References

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The prescribed text for this subject is a preliminary version of the book

J. J. Koliha, Linear Analysis, University of Melbourne, 20065

sold from the University Bookroom. The following books may serve as additional references:

I. Integration

• Serge Lang, Real and Functional Analysis, Springer 1993
• G. B. Bartle, The Elements of Integration and Lebesgue Measure, Wiley 1995
• W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill 1987
• C. Swartz, Measure, Integration and Function Spaces, World Scientific 1994
• (Historical interest) Henri Lebesgue, Lecons sur l'integration et la recherche des fonctions primitive, Gauthier-Villars 1904
• (Historical interest) Henri Lebesgue, Measure and the Integral (Edited with a biographical essay by Kenneth O. May), Holden-Day, San Francisco,1966 (available in Baillieu Library 517.3 L443)

II. Linear spaces and operators

• A. L. Brown and A. Page, Elements of Functional Analysis, Van Nostrand Reinhold 1970
• E. Kreyszig, Functional Analysis with Applications, Wiley 1978
• (Historical interest) S. Banach, Theorie des Operations Lineaires, Warsaw 1932
• F. Riesz and B. Sz.-Nagy, Functional Analysis, F. Ungar 1955
• L. Debnath and P. Mikusinski, Introduction to Hilbert Spaces with Applications, Academic Press 1999 (interesting applications of Fourier series and introduction to wavelets)
• P. R. Halmos, A Hilbert Space Problems Book, Van Nostrand, Princeton 1967

Practice Class

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Practice Class is given weekly on Thursday 10-11 am in Room 213 in R. Berry Building. Attendance is monitored. Overall strategy for Practice Classes:

• Attempt problems beforehand, and prepare questions you want to ask. Coming in cold is not nearly as effective. The results of certain problems will be used later, so even if you do not solve them, you must know what they say.
  
  
• Bring the printed Lecture Notes as well as your own hand written notes. Your lecturer suggests that you should keep a list of main definitions and theorems, enlarging it as you go. This is an invaluable help for the swat week.
  
  
  PROBLEMS to concentrate on:
  
  
• Section 12.2: 6, 8, 10, 11, 12, 14, 15, 19, 20, 23
  
  
• Section 13.7: 1, 3, 6, 7, 8, 12, 13, 14, 15, 19, 20, 21, 27, 28, 29, 30 (compare 14 with 29, and 15 with 30), a selection from 31-36
  
  
• Section 14.3: a selection from 1-12, then 13-15, 19, a selection of cases from 21; then 23 and 24
  
  
• Section 15.6: 1, 2, 6 (a selection), 10 (not every Stieltjes gauge on R^k arises in this way), 12, 13, 16, 17, 19
  
  
• Section 16.4: 2, 8 (a selection), 9-11, 13, 15 (a selection)
  
  
• Section 7.4: all problems except 19 and 20
  
  
• Section 8.3: 16, 17 20, 23, 27 (optional 26)
  
  
• Section 9.6: 1-10 (in particular 5 and 9), a selection from 11; then 13, 18, 20, 28, 40-43
  
  

Subject Objectives

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Handbook entry for 620-312

Content:

Measure and integration: Introduction to measure spaces and abstract Lebesgue integration; Lebesgue integration in Euclidean spaces; dominated convergence and applications; relation between integration and differentiation; Newton integral

Linear spaces and operators: Normed and inner product spaces; Hilbert spaces, abstract Fourier series; linear operators and functionals; dual spaces; Hahn-Banach theorem; uniform boundedness; open mapping theorem and closed graph theorem

On completion of this subject, students should:

Comprehend:

• fundamental concepts about measure and Lebesgue integration with respect to a variety of measures;
• how basic concepts of linear algebra can be generalised to infinite dimensional situation using techniques from analysis;
• thow these concepts arise in many branches of mathematics, as for example, in partial differential equations, operations research and probability, but also in areas of theoretical physics such as quantum mechanics;

Have developed the ability to:

• give rigorous mathematical arguments at an advanced level;
• apply abstract concepts to solving problems in other areas of mathematics such as differential equations;

Appreciate:

• the necessity for a rigorous theoretical foundation of concepts used frequently in mathematics and physics.
  
  

Generic Skills

In addition to learning specific technical skills that will assist you in your future careers in science, engineering, commerce, education or elsewhere, you will have the opportunity to develop in this subject, generic skills that will assist you whatever your future career path.

• You will develop problem-solving skills including engaging with unfamiliar problems, and identifying relevant strategies.
  
  
• You will develop analytical skills - the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of the analysis.
  
  
• Through interactions with fellow students, you will develop the ability to work in a team. The department distinguishes between ethical collaboration, which is strongly encouraged, and plagiarism, which is prohibited.
  
  
• You will develop your oral presentation skills, practicing presentation of technical solutions. This practice will assist you in learning how to present material in a well-organized, well-structured, lucid and persuasive fashion.
  
  
• With both team-based and individually assessable material to be submitted throughout the semester, you will learn to manage your time, balance competing commitments and set and meet regular deadlines; this is formalized through the project scope document.
  
  
  

©The University of Melbourne 1994-1999. Disclaimer and Copyright Information.

Created: July 2001
Last modified:11 October 2007
Authorised by: Professor Peter Taylor, Head of Department