MATH4FA Functional Analysis
The main focus of this course is the analysis of linear mappings between normed linear spaces. It turns out that many problems in analysis can be studied from this abstract point of view, which recognizes important underlying principles without getting lost in technical details. The applications of the approach ranges from differential and integral equations through problems in optimal control theory and numerical analysis to probability theory to name a few. This introductory course covers some of the basic constructions and principles of functional analysis: completions of metric/normed spaces, the Hahn-Banach Theorem and its consequences, dual spaces, bounded linear operators and their adjoints, closed operators, the Open Mapping and Closed Graph Theorems, the Principle of Uniform Boundedness and compactness in metric spaces. The applications of the abstract concepts are demonstrated through various examples from different branches of analysis.
2018, Semester 1.
MATH302 Complex Analysis, with the honours module "MATH4MI Measure and Integration" highly recommended.
Markus Antoni (Room 310a, phone 479 4567, email: firstname.lastname@example.org)
There are only 7 weeks of teaching (weeks 10-13,15,18,19). Lectures will be on
Mondays, 10am -11am, room 241 (9am - 11am, room 241, on weeks 13 and 19)
Tuesdays, 2pm - 3pm, room 240
Wednesdays, 12pm - 1pm, room 240
J.B. Conway, A Course in Functional Analysis, Springer, 1990.
W. Rudin, Functional Analysis, McGraw-Hill, 1991.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
N. Dunford and J.T. Schwartz, Linear Operators. Part I. General Theory, John Wiley & Sons, 1988.
M. Reed and B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis, Academic Press, 1972.
M. Schechter, Principles of Functional Analysis, American Mathematical Society, 2002.
E.M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, 2nd Print, Princeton University Press, 2011.
A.E. Taylor and D.C. Lay, Introduction to Functional Analysis, Robert E. Krieger Publishing Co., 1986.
There will be three assignments. The first one will consist of 10 exercises (8 will be marked), the second one of 5 exercises (4 will be marked), and the third one again of 5 exercises (3 will be marked). In addition to that there will be weekly homework sheets just for practice.
The final exam will take place on Monday, May 28, 9am - 12pm, in room 240.
Your final mark F in the paper will be calculated according to this formula:
F = max(E, (0.5E + 0.5A))
- E is the Exam mark
- A is the Assignments mark
and all quantities are expressed as percentages.
Students must abide by the University’s Academic Integrity Policy
Academic integrity means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.
Academic misconduct is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.
Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.
All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.
Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.
If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.
Types of academic misconduct are as follows:
The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).
- Although not intended, unintentional plagiarism is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided.
- Intentional plagiarism is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.
Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..
Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.
Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.
Use of Unauthorised Materials
Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.
Assisting Others to Commit Academic Misconduct
This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.