## PURE MTH 3002 - Topology and Analysis III

### North Terrace Campus - Semester 1 - 2015

Solving equations is a crucial aspect of working in mathematics, physics, engineering, and many other fields. These equations might be straightforward algebraic statements, or complicated systems of differential equations, but there are some fundamental questions common to all of these settings: does a solution exist? If so, is it unique? And if we know of the existence of some specific solution, how do we determine it explicitly or as accurately as possible? This course develops the foundations required to rigorously establish the existence of solutions to various equations, thereby laying the basis for the study of such solutions. Through an understanding of the foundations of analysis, we obtain insight critical in numerous areas of application, such areas ranging across physics, engineering, economics and finance. Topics covered are: sets, functions, metric spaces and normed linear spaces, compactness, connectedness, and completeness. Banach fixed point theorem and applications, uniform continuity and convergence. General topological spaces, generating topologies, topological invariants, quotient spaces. Introduction to Hilbert spaces and bounded operators on Hilbert spaces.

• General Course Information
##### Course Details
Course Code PURE MTH 3002 Topology and Analysis III Pure Mathematics Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week Y MATHS 2100 MATHS 2100 Solving equations is a crucial aspect of working in mathematics, physics, engineering, and many other fields. These equations might be straightforward algebraic statements, or complicated systems of differential equations, but there are some fundamental questions common to all of these settings: does a solution exist? If so, is it unique? And if we know of the existence of some specific solution, how do we determine it explicitly or as accurately as possible? This course develops the foundations required to rigorously establish the existence of solutions to various equations, thereby laying the basis for the study of such solutions. Through an understanding of the foundations of analysis, we obtain insight critical in numerous areas of application, such areas ranging across physics, engineering, economics and finance. Topics covered are: sets, functions, metric spaces and normed linear spaces, compactness, connectedness, and completeness. Banach fixed point theorem and applications, uniform continuity and convergence. General topological spaces, generating topologies, topological invariants, quotient spaces. Introduction to Hilbert spaces and bounded operators on Hilbert spaces.
##### Course Staff

Course Coordinator: Professor Finnur Larusson

##### Course Timetable

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

• Learning Outcomes
##### Course Learning Outcomes
1. Demonstrate an understanding of the concepts of metric spaces and topological spaces, and their role in mathematics.
2. Demonstrate familiarity with a range of examples of these structures.
3. Prove basic results about completeness, compactness, connectedness and convergence within these structures.
4. Use the Banach fixed point theorem to demonstrate the existence and uniqueness of solutions to differential equations.
5. Demonstrate an understanding of the concepts of Hilbert spaces and Banach spaces, and their role in mathematics.
6. Demonstrate familiarity with a range of examples of these structures.
7. Prove basic results about Hilbert spaces and Banach spaces and operators between such spaces.
8. Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
9. Demonstrate skills in communicating mathematics orally and in writing.
##### University Graduate Attributes

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

University Graduate Attribute Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. 1,2,3,4,5,6,7
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 3,4,7,8
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 8
Skills of a high order in interpersonal understanding, teamwork and communication. 9
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all
A commitment to the highest standards of professional endeavour and the ability to take a leadership role in the community. all
• Learning Resources
None.
##### Recommended Resources
• Cohen, Graham, "A course in modern analysis and its applications"
• Simmons, George F., "Introduction to topology and modern analysis''
• Apostol, Tom M., "Mathematical analysis''
• Kreyszig, Erwin,  "Introductory functional analysis with applications''
• Sutherland, Wilson A., "Introduction to metric and topological spaces''
• Munkres, James, "Topology"
• Larusson, Finnur, "Lectures on real analysis" (the last two chapters)
##### Online Learning

Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.
• Learning & Teaching Activities
##### Learning & Teaching Modes
The lecturer will guide the students through the course material in 30 lectures. Students are expected to actively engage with the material during the lectures, and interaction and discussion of any difficulties that arise during the lectures is encouraged. Students are expected to attend all lectures, but (where possible) the lectures will be recorded to help cover absences and for revision purposes. Students will be expected to present solutions to fortnightly tutorial problems. Fortnightly assignments help develop understanding of the theory and its applications, and timely feedback allows students to gauge their progress.
##### Workload

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 Tutorials 5 25 Assignments 6 42 Total 157

##### Learning Activities Summary

 Week 1 Metric spaces Metric spaces, examples, convergent sequences, open and closed sets. Week 2 Metric spaces Open and closed sets (cont.), Cauchy sequences, complete metric spaces. Week 3 Metric spaces Continuous maps, the Banach fixed point theorem, motivation and examples. Week 4 Metric spaces Picard's existence and uniqueness theorem for solutions of differential equations. Week 5 Metric spaces Compactness, uniform continuity, the Heine-Borel theorem, the Arzela-Ascoli theorem. Week 6 Topology Topological spaces, examples, Hausdorff spaces, compact spaces. Week 7 Topology Continuous maps, homeomorphisms, connected and path connected spaces. Week 8 Topology Connected and path connected spaces (cont.).  Normed vector spaces, Banach spaces, examples. Week 9 Hilbert and Banach spaces Bounded linear maps, bounded linear functionals, dual spaces. Week 10 Hilbert and Banach spaces Inner products, Cauchy-Schwarz inequality, parallellogram law, orthogonality. Week 11 Hilbert and Banach spaces Hilbert spaces, examples, orthogonal projections, Riesz representation theorem. Week 12 Hilbert and Banach spaces Adjoint operators, structure theorem for separable Hilbert spaces.
Tutorials in Weeks 3, 5, 7, 9, 11 cover the material of the previous two weeks.
• 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

Assessment taskTask typeDueWeightingLearning outcomes
Examination Summative Examination period 70% All
Homework assignments Formative and summative Weeks 2, 4, 6, 8, 10, 12 25% All
Tutorial exercises Formative Weeks 3, 5, 7, 9, 11 5% All
##### Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
##### Assessment Detail

Assessment taskSetDueWeighting
Assignment 1 Week 1 Week 2 4 1/6%
Tutorial exercises 1 Week 2 Week 3 see below
Assignment 2 Week 3 Week 4 4 1/6%
Tutorial exercises 2 Week 4 Week 5
Assignment 3 Week 5 Week 6 4 1/6%
Tutorial exercises 3 Week 6 Week 7
Assignment 4 Week 7 Week 8 4 1/6%
Tutorial exercises 4 Week 8 Week 9
Assignment 5 Week 9 Week 10 4 1/6%
Tutorial exercises 5 Week 10 Week 11
Assignment 6 Week 11 Week 12 4 1/6%

Each student will present at least once in the tutorials. Tutorial presentations will be worth 5%. This may have to be adjusted depending on enrolment.
##### Submission
Homework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons.  Documentation is required and the lecturer must be notified as soon as possible.
##### Course Grading

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

M10 (Coursework Mark Scheme)
Grade Mark Description
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

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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