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.
None. MATH302 is highly recommended.
Markus Antoni (Room 310a, phone 479 4567, email: firstname.lastname@example.org)
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.
Your final mark F in the paper will be calculated according to this formula:
F = max(E, (2E + A)/3)
- 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 endeavours at the University of Otago are built upon an essential commitment to academic integrity.
The two most common forms of academic misconduct are plagiarism and unauthorised collaboration.
Academic misconduct: Plagiarism
Plagiarism is defined as:
- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.
All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.
Academic misconduct: Unauthorised Collaboration
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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.