PURE MTH 4012 - Pure Mathematics Topic B - Honours

North Terrace Campus - Semester 1 - 2016

Please contact the School of Mathematical Sciences for further details, or view course information on the School of Mathematical Sciences web site at http://www.maths.adelaide.edu.au

• General Course Information
Course Details
Course Code PURE MTH 4012 Pure Mathematics Topic B - Honours School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Y Please contact the School of Mathematical Sciences for further details, or view course information on the School of Mathematical Sciences web site at http://www.maths.adelaide.edu.au
Course Staff

Course Coordinator: Steve Rosenberg

Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
Course Learning Outcomes
In 2016, the topic of this course is FUNCTIONAL ANALYSIS.

Syllabus

In general, functional analysis can be defined as the study of infinite dimensional vector spaces and operators on these spaces. Functional analysis as a separate mathematical subject emerged about a century ago, motivated in part by the success of using Hilbert spaces to rigorously justify Fourier series expansions for functions on the unit circle. In the extension of this theory to other settings, fundamental questions in functional analysis appear: (i) What are the appropriate topologies to put on the infinite dimensional vector spaces that appear as spaces of functions? (ii) To what extent do linear differential operators acting on spaces of functions (like d/d theta acting on functions on the circle) behave like linear transformations on finite dimensional vector spaces?

The first question is highly nontrivial and depends on the context, precisely because infinite dimensional vector spaces have many inequivalent norm topologies. Investigating the second question quickly leads to the realization that even the simplest linear differential operator like d/d theta is discontinuous in the most reasonable topologies. On the other hand, standard operators like Green's operators (e.g. (d/d theta)^{-2}, roughly speaking the inverse of the Laplacian on the circle) are continuous in these topologies. Therefore, functional analysis naturally splits into the study of continuous operators on function spaces and discontinuous operators - both are important.

More specifically, this course will cover the basic topologies on infinite dimensional vector spaces, including Hilbert spaces (inner product topologies), Banach spaces (norm topologies), and Frechet spaces (topologies built to handle spaces of smooth functions). Specific examples will include L^p and Sobolev spaces. We will restrict attention to continuous linear operators between these spaces, with applications to the study of differential operators. We will discuss the spectral theorem about diagonalization of linear operators on Hilbert spaces with Fourier series as the guiding example. As time permits, we'll discuss more advanced results like the Hahn-Banach theorem and the theory of distributions, the rigorous treatment of delta functions.

Assumed knowledge: Topology and Analysis III.

Learning Outcomes

On successful completion of this course, students will be able to

1. Define complete normed spaces, and understand the basic examples of Banach and Hilbert spaces;
2. Define Frechet spaces and understand the basic examples;
3. Understand the spectral theorem for compact selfadjoint operators on Hilbert spaces, with basic examples;
4. Understand the Hahn-Banach theorem, open mapping theorem and closed graph theorem;
5. Understand the basics of distribution theory.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
all
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
all
• Learning Resources
None.
Recommended Resources

1. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations"

2. Conway, "A Course in Functional Analysis"

3. Kreyszig, "Functional Analysis With Applications"

4. Riesz and Nagy, "Functional Analysis"

5. Rudin, "Functional Analysis"
Online Learning
The course will have an active MyUni website.
• Learning & Teaching Activities
Learning & Teaching Modes
The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload hours Lectures 30 90 Assignments 6 66 Total 156
Learning Activities Summary

No information currently available.

• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

Assessment Summary
Examination Summative Examination period 70% All
Homework assignments Formative and summative One week after assigned 30% All
Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
Assessment Detail
There will be a total of 6 homework assignments, due one week after assigned. Each will cover material from the lectures, and in addition, will sometimes go beyond that so that students may have to undertake some additional research.
Submission
Homework assignments must be given to the lecturer in person or emailed as a pdf. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M11 (Honours Mark Scheme)
GradeGrade reflects following criteria for allocation of gradeReported on Official Transcript
Fail A mark between 1-49 F
Third Class A mark between 50-59 3
Second Class Div B A mark between 60-69 2B
Second Class Div A A mark between 70-79 2A
First Class A mark between 80-100 1
Result Pending An interim result RP
Continuing Continuing CN

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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