Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

MATH4FA Functional Analysis

Second Semester
10 points

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.

Prerequisites

MATH302 Complex Analysis, with the honours module "MATH4MI Measure and Integration" highly recommended

Lecturer

Markus Antoni (Room 310a, phone 479 4567, email: mantoni@maths.otago.ac.nz)

First meeting

There will be an introductory meeting on Tuesday, July 11, at 11 am in room 240.

Lectures

Mondays, 10 am - 12 pm, room 240.

Literature

Recommended reading

J.B. Conway, A Course in Functional Analysis, Springer, 1990.

W. Rudin, Functional Analysis, McGraw-Hill, 1991.

Further literature

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.

Internal assessment

There will be weekly written assignments consisting of two exercises that will be marked and some others just for practice. You are only required to hand in the exercises to be marked. However, you are strongly encouraged to work on the other exercises, too.

Final exam

The final exam will take place on Wednesday, November 8, 10am - 1pm, in room 240.

Final mark

Your final mark F in the paper will be calculated according to this formula:

F = max(E, 0.5A + 0.5E)

where:

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.

Further information

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.