## MATHS 7100 - Real Analysis

### North Terrace Campus - Semester 2 - 2015

Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called "analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered are: Basic set theory. The real numbers and their basic properties. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions and uniform continuity. The Riemann integral. Differentiation and Mean Value theorems. The Fundamental Theorem of Calculus. Series. Power series and Taylor series. Convergence of sequences and series of functions.

• General Course Information
##### Course Details
Course Code MATHS 7100 Real Analysis School of Mathematical Sciences Semester 2 Postgraduate Coursework North Terrace Campus 3 Up to 3.5 hours per week Y MATHS 1012 Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called "analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered are: Basic set theory. The real numbers and their basic properties. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions and uniform continuity. The Riemann integral. Differentiation and Mean Value theorems. The Fundamental Theorem of Calculus. Series. Power series and Taylor series. Convergence of sequences and series of functions.
##### Course Staff

Course Coordinator: Associate Professor Nicholas Buchdahl

##### Course Timetable

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

• Learning Outcomes
##### Course Learning Outcomes
On successful completion of this course, students will be able to

1. describe fundamental properties of the real numbers that lead to the formal development of real analysis;

2. comprehend rigorous arguments developing the theory underpinning real analysis;

3. demonstrate an understanding of limits and how they are used in sequences, series, differentiation and integration;

4. construct rigorous mathematical proofs of basic results in real analysis;

5. appreciate how abstract ideas and rigorous methods in mathematical analysis can be applied to important practical problems.

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. all
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 1,3,4
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 2,4,5
Skills of a high order in interpersonal understanding, teamwork and communication. 4
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 5
• Learning Resources
None.
##### Recommended Resources
There are many books on real analysis available in the library. Amongst those, the following is a selection of some that are very compatible with the level and objectives of this course:

1.  Belding & Mitchell: "Foundations of Analysis";
2.  Fitzpatrick: "Real Analysis";
3.  Gaughan: "Introduction to Analysis";
4.  Lárusson: "Lectures on Real Analysis".
##### Online Learning
Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.

• Learning & Teaching Activities
##### Learning & Teaching Modes
This course relies on lectures as the primary delivery mechanism for the material. Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through lectures. A sequence of written assignments provides assessment opportunities for students to gauge their progress and understanding.

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

 Activity Quantity Workload hours Lectures 36 90 Tutorials 6 18 Assignments 6 42 Test 1 8 TOTAL 158
##### Learning Activities Summary
Lecture Outline

Basic logic and set theory. The real numbers and their defining properties. (Lectures 1-4)
Sequences: convergence, properties of limits, subsequences. (Lectures 5-8)
The topology of the real numbers. (Lectures 9-11)
Continuity. The key properties of continuous real-valued functions of a real variable. (Lectures 12-14)
The Riemann integral. (Lectures 16-19)
Differentiation. Mean Value theorems, l'Hôpital's rules. (Lectures 20-22)
The inverse function theorem. The fundamental theorem of calculus. (Lectures 23-25)
Series. Tests for convergence. (Lectures 26-29)
Power series. Taylor's theorem. Sequences and series of functions. (Lectures 30-33)
Convergence of sequences and series of functions. (Lectures 34-36)

Tutorials

Tutorials will be held in every odd week, covering material from the preceding two weeks of lectures. The first tutorial, in Week 1, will be a review of relevant ideas from Mathematics IA and IB Calculus.
• 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

 Component Task type Due Weighting Objectiveassessed Tutorial attendance Formative Odd weeks 2 All Tutorial performance Formative Odd weeks 3 All Assignments Formative andsummative Odd weeks 20 All Test Summative Lecture 15 20 1,2,3,4 Exam Summative Exam period 55 All
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail
Assignments will be distributed in the middle of each odd week and due at the end of each even week. Each will mainly cover material from the previous weeks' lectures, but may include material from the next one or two lectures. Of the six assignments, the scores from the best five together with half of the worst will be used to calculate the final mark for assignments. In all, assignments count 20% towards the final mark for the course.

The midsemester test will be held during the 15th lecture period. It will cover the material covered in lectures up to the end of Lecture 12. It will be worth 20% of the final mark.
##### Submission
All written assignments are to be submitted to the designated hand-in boxes in the School of Mathematical Sciences. Late assignments will not be accepted.

Assignments will have a two week turn-around time for feedback to students.

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

M10 (Coursework Mark Scheme)
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
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