Analysis 1 is a foundational course in Mathematics, leading on to other areas of analysis, such as topology and measure theory, complex analysis, functional analysis, and harmonic analysis. It also provides important tools for application areas such as theoretical computer science, physics and engineering.

Topics to be covered include: Elementary set theory; metric spaces, sequences, uniform convergence, continuity, the contraction mapping principle; integral equations and differential equations, topological spaces.

Note: This course will have shared lectures with MATH3116 and MATH6110.

Note: This is an Honours Pathway Course: It emphasises mathematical rigor and proof and develops the application of the theory to topics such as differential equations.

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

1. Explain the fundamental concepts of real analysis and their role in modern mathematics and applied contexts.
2. Demonstrate accurate and efficient use of real analysis techniques.
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from real analysis.
4. Apply problem-solving using real analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.

Examination Material or equipment

Any exams

Required Resources

Lecture notes can be downloaded from the course Wattle site. Almost everything we cover in class will be in these notes. Other material will also be provided in Wattle, e.g. additional notes, alternative proofs, etc.

Recommended Resources

There are many excellent texts on the topics we are covering. Most have a title like ‘Introduction to analysis’ or ‘Introduction to metric spaces’ or similar. Finding a book you like and doing problems from it is a good way to learn the material.

Some suggested books are:

• W. Rudin, Principles of Mathematical Analysis, McGraw-Hill. (A bit difficult, but good.)
• J. Giles, Introduction to the Analysis of Metric Spaces, Australian Mathematical Society lecture series no. 3. (Almost exactly right for the metric space component.)
• G. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill. (At a higher level and deals with much more material, though the book will serve you well in later courses.)
• H. Royden, Real Analysis, Prentice-Hall, various editions. (Like Simmons, this does more than you need but is good for several later courses.) The latest edition is by Royden and Fitzpatrick; it includes more material than the Royden editions.
• T.W. Korner, A Companion to Analysis, American Mathematical Society. (A different kind of textbook, designed to help you think about the material.)

There is also a free text by W. Trench available on the course Wattle site. It is not perfectly suited to the course, but covers some of the same material.

This course is, in the most part, delivered in-person, on-campus. Whether you are on campus or studying online, there are a variety of online platforms you will use to participate in your study program. These could include videos for lectures and other instruction, two-way video conferencing for interactive learning, email and other messaging tools for communication, interactive web apps for formative and collaborative activities, print and/or photo/scan for handwritten work and drawings, and home-based assessment.

ANU outlines recommended student system requirements to ensure you are able to participate fully in your learning. Other information is also available about the various Learning Platforms you may use.

Staff Feedback

Students will be given feedback in the following forms in this course:

• written comments
• verbal comments
• feedback to whole class, groups, individuals, focus group etc

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). Feedback can also be provided to Course Conveners and teachers via the Student Experience of Learning & Teaching (SELT) feedback program. SELT surveys are confidential and also provide the Colleges and ANU Executive with opportunities to recognise excellent teaching, and opportunities for improvement.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Content block 1. Set theory: We will discuss some of the main ideas in set theory, but this will be an overview of the topic rather than an in depth examination.	For the whole course: 3 lectures per week; 1 workshop in most weeks; 5 assignments throughout the semester; a mid-semester exam; and a final examination.
2	Content block 2. Metric spaces: The central part of the course is the analysis of metric spaces, that is, sets that have a notion of distance between points. Much of what we will do involves looking at important examples from various parts of mathematics. We will apply metric space theory to several different problems: the existence of solutions of ordinary differential equations, the existence and construction of fractal sets, and the inverse and implicit function theorems, are some that are included in the notes.	See above.
3	Content block 3. Topological spaces: Topological spaces are a natural generalisation of metric spaces. We can define concepts such as convergence, compactness, continuity and connectedness in topological spaces without requiring a notion of distance. We will cover some basic theory of topological spaces in the last two to three weeks of the course. Most spaces that arise in further analysis courses are metric spaces, but there are some important examples of topological spaces that are not metric spaces.	See above.

Tutorial Registration

Workshops begin in Week 3. Workshop registration is via MyTimetable. ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities / tutorials so they can better plan their time. Find out more on the Timetable webpage.

Assessment Summary

Assessment task	Value	Learning Outcomes
Assignments	25 %	1,2,3,4
Mid-semester Exam	30 %	1,2,3,4
Final Exam	45 %	1,2,3,4

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines , which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Integrity Rule before the commencement of their course. Other key policies and guidelines include:

• Academic Integrity Policy and Procedure
• Student Assessment (Coursework) Policy and Procedure
• Extenuating Circumstances Application
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure
• Code of practice for teaching and learning

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the Academic Skills website. In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Participation

In Semester 1 2024, this course is delivered in-person, on campus.

Examination(s)

This course includes a mid-semester and a final examination. The details and mode of delivery for exams will be communicated through the course Wattle site and the ANU examination timetable. It is likely that the mid-semester exam will be held online, and the final exam in-person. There is a hurdle requirement on the final exam: to pass the course, a student must score at least 34 on the final exam.

Assessment Task 1

Assignments

There will be five assignments throughout the semester, worth a total of 25% of the final grade::

• Assignment 1: Available by Monday 26 February, due Tuesday 12 March.
• Assignment 2: Available by Tuesday 12 March, due Tuesday 26 March.
• Assignment 3: Available by Tuesday 26 March, due Monday 22 April.
• Assignment 4: Available by Monday 22 April, due Tuesday 7 May.
• Assignment 5: Available by Tuesday 7 May, due Friday 24 May.

Assessment Task 2

Mid-semester Exam

To be held in Week 6. Worth 30% of the final grade. You may refer to the course lecture notes during the exam, however other aids or sources of information are not permitted.

Check the course Wattle site for more information and to confirm the date, time and mode of the exam.

Assessment Task 3

Final Exam

To be held during the final examination period. Worth 45% of the final grade. Check the Course Wattle site for further information, and the Central ANU examination timetable to confirm the date, time and location of the exam.

There is a hurdle requirement on the final exam: to pass the course, a student must score at least 34 on the final exam.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. The University’s students are an integral part of that community. The academic integrity principle commits all students to engage in academic work in ways that are consistent with, and actively support, academic integrity, and to uphold this commitment by behaving honestly, responsibly and ethically, and with respect and fairness, in scholarly practice.

The University expects all staff and students to be familiar with the academic integrity principle, the Academic Integrity Rule 2021, the Policy: Student Academic Integrity and Procedure: Student Academic Integrity, and to uphold high standards of academic integrity to ensure the quality and value of our qualifications.

The Academic Integrity Rule 2021 is a legal document that the University uses to promote academic integrity, and manage breaches of the academic integrity principle. The Policy and Procedure support the Rule by outlining overarching principles, responsibilities and processes. The Academic Integrity Rule 2021 commences on 1 December 2021 and applies to courses commencing on or after that date, as well as to research conduct occurring on or after that date. Prior to this, the Academic Misconduct Rule 2015 applies.

 

The University commits to assisting all students to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. All coursework students must complete the online Academic Integrity Module (Epigeum), and Higher Degree Research (HDR) students are required to complete research integrity training. The Academic Integrity website provides information about services available to assist students with their assignments, examinations and other learning activities, as well as understanding and upholding academic integrity.

Online Submission

Assignment submission, and any remotely invigilated exam submission, will be online, via a process that will be detailed on the course Wattle page. You will be required to electronically sign a declaration as part of the submission of your assignment. Please keep a copy of the assignment for your records. This course does not use Turnitin, having been granted an exemption.

Hardcopy Submission

It is expected that all assessment submission for this course will be online (other than for the in-person final exam). For some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.

Late Submission

Late submission not permitted. Submission of assessment tasks without an extension after the due date is not permitted: a mark of 0 will be awarded.

Referencing Requirements

The Academic Skills website has information to assist you with your writing and assessments. The website includes information about Academic Integrity including referencing requirements for different disciplines. There is also information on Plagiarism and different ways to use source material.

Returning Assignments

Graded assignments will be returned electronically.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

Resubmission of assignments is not permitted.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes.

Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Access and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents all ANU students