# 35004 Mathematical Analysis and Applications

**Warning:** The information on this page is indicative. The subject outline for a
particular session, location and mode of offering is the authoritative source
of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.

Subject handbook information prior to 2024 is available in the Archives.

*UTS: Science: Mathematical and Physical Sciences*

*Credit points:*8 cp

*Result type*: Grade and marks

*There are course requisites for this subject. See access conditions.*

### Description

This subject is an introduction to the area of Functional Analysis, one of the most important and powerful mathematical areas developed over the last one hundred years, and the natural extension of mathematical analysis. Students study Banach spaces and Hilbert spaces, infinite dimensional normed vector spaces. They study the major results including the Closed Graph Theorem, the Hahn Banach Theorem and the Spectral Theorem for self-adjoint Operators. Students also study applications to the solutions of Partial Differential Equations, the Theory of Distributions and the theory of Wavelets.

The tools we develop have found major applications in mathematical physics, quantitaive finance and mathematical biology.

### Subject learning objectives (SLOs)

Upon successful completion of this subject students should be able to:

1. | Elaboratively build one's mathematical language and arguments to explain definitions and theorems relevant to the subject, presenting the information with the highest standards of rigour and professionalism |
---|---|

2. | Demonstrate how language is used when referencing definitions and theorems at a conceptual level to one's peers |

3. | Analyse and synthesise the methods of proof of the theorems and be able to prove these theorems. Validate and test ideas in the theory of the integral, Fourier analysis, functional analysis and distribution theory. |

4. | Hypothesise and prove or disprove results the students have not seen before, using methods of advanced mathematical analysis. |

5. | Apply results from the subject to problems arising in related disciplines including probability theory and engineering. |

### Course intended learning outcomes (CILOs)

This subject also contributes specifically to the development of following course intended learning outcomes:

- Demonstrate appraisal and interpretation of mathematical and statistical principles and concepts, with a focus on applying methodologies to solve complex problems. (1.1)
- Tackle the challenge of complex real-world problems in the areas of mathematical and statistical modelling by evaluating and analysing information and solutions. (2.1)
- Engage in work practices that demonstrate an understanding of confidentiality requirements, ethical conducts, data management, and organisation and collaborative skills in the context of applying mathematical and statistical modelling. (3.1)
- Identify, present complex ideas and justifications using appropriate communication approaches from a variety of methods (oral, written, visual) to communicate with mathematicians, data analysts, scientists, industry, and the general public. (5.1)

### Contribution to the development of graduate attributes

This subject is expected to contribute to the following honours level graduate profile attributes:

**1 . Disciplinary Knowledge**

The subject introduces the students to the advanced mathematical tools and ideas that are essential to modern mathematics. The material they learn lies at the very foundations of modern mathematics and an enormous range of applications. For example, modern signal processing, which is essential to mobile phone technology, is based on Fourier analysis, which the students learn in this subject. This is assessed in all assessment tasks.

**2. Research, Inquiry and Critical Thinking**

In the subject students learn how new results and areas of research arise from actual problems. For example the modern theory of integration, which is covered in the subject, developed because the existing theory proved inadequate for the solution of problems in Fourier analysis. The student will be asked to prove results that they have not seen before using the methods developed in the subject. This is developed and assessed across the whole subject in all assessments.

**3. Professional, Ethical and Social Responsibility**

The ability to acquire, develop, employ and integrate a range of technical, practical and professional skills, in appropriate and ethical ways within a professional context, autonomously and collaboratively and across a range of disciplinary and professional areas. Students will be expected to take an unfamiliar problem, analyse it, formulate a solution and present this in a clear and professional manner. This attribute is developed throughout the subject, particuarly in the assignments, which involve advanced problem solving skills.

**5. Communication **

Students learn to communicate mathematical ideas and results for an audience of peers and the wider mathematical community. This attribute is developed and assessed throughout the subject, particularly in the assignments.

### Teaching and learning strategies

*The material online will be discussed actively with students strongly encouraged to attend and participate, as this will benefit their performance in the subject. Online attendance will be possible if students are not able to attend for some reason.*

Canvas is used extensively. A complete set of class notes will be posted, as well as class problems with solutions and assignments. In addition, informal notes will be posted to stimulate class discussion.

Students are strongly encouraged to prepare for each class by doing the problem sets and reading material from the notes.

Students are able to collaborate with their peers on the solution of the weekly problem sets. Students may be asked to present solutions to problems to the class.

Students are given regular written and verbal feedback in class. The most common errors being made by the students are addressed in class with explanations and advice on how to avoid them.

There will be three assessment tasks, two assignments due in weeks 5 and 10 and a final assignment in week 12. Feedback is given on assignments when they are returned to the students.

The assessment tasks are based upon the application of the theory learned in class and through other resources, such as the online notes, which provide greater depth than can be covered in a two hour lecture. Reading recommended texts and doing exercises from the texts plays an essential part in the study of this subject.

In general, students should expect to spend 6 hours per week on this subject outside class time.

### Content (topics)

**Module 1 Topological Vector Spaces**

This module starts with a review of metric spaces and Lebesgue Integration, and then moves on to the notion of Topological Vector Spaces including Hilbert Spaces and Banach Spaces, continuous linear mappings, local convexity. We study completeness, compactness, the Open Mapping theorem, the Hahn Banach Theorem, and the Krein-Milman Theorem

**Module 2 Duality, Distributions and the Spectral Theorem**

We study the notion of the dual of a Banach space, the notion of an adjoint operator, and then move on to study the theory of distributions, their Fourier transforms and applications to Partial Differential Equations.

**Module 3 Bounded and Unbounded operators and Banach Algebras**

In this module, we look at Banach algebras and the spectral theorem, resolutions of the identity, B* and C* algebras, and the spectral theory of unbounded operators. We finish the course with a quick introduction to Frames and Wavelet bases.

### Assessment

#### Assessment task 1: Assignment 1

Intent: | This assessment contributes to the following Graduate attributes 1. Disciplinary knowledge 2. Research, Inquiry and Critical Thinking 3. Professional, Ethical and Social Responsibility 5. Communication |
---|---|

Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1 and 5.1 |

Type: | Exercises |

Groupwork: | Group, individually assessed |

Weight: | 25% |

Criteria: | Accuracy of proofs and calculations, Clarity of answers, Correctness of results, insight and creativity, communication of ideas |

#### Assessment task 2: Assignment 2

Intent: | This assessment contributes to the following Graduate attributes 1. Disciplinary knowledge 2. Research, Inquiry and Critical Thinking 3. Professional, Ethical and Social Responsibility 5. Communication |
---|---|

Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1 and 5.1 |

Type: | Exercises |

Groupwork: | Group, individually assessed |

Weight: | 25% |

Criteria: | Accuracy of proofs and calculations, Clarity of answers, Correctness of results, insight and creativity, communication of ideas |

#### Assessment task 3: Assignment 3

Intent: | This assessment contributes to the following Graduate attributes 1. Disciplinary knowledge 2. Research, Inquiry and Critical Thinking 5. Communication |
---|---|

Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1 and 5.1 |

Type: | Exercises |

Groupwork: | Group, individually assessed |

Weight: | 50% |

Criteria: | Accuracy of proofs and calculations, Clarity of answers, Correctness of results, insight and creativity, communication of ideas |

### Minimum requirements

The final grade will be F = A1+A2+A3, where A1 and A2 are the marks for the first two assignments out of 25 and A3 is the mark for assignment three out of 50.

### Recommended texts

The following textbooks are useful, but not required.

- Capinski, M. and Kopp, E. Measure, Integral and Probability, 2nd Edition., Springer 2005.
- Robdera, M. A. A Concise Approach to Mathematical Analysis, Springer 2003.
- Cohen, G. l. A Course in Modern Analysis and its Applications. Cambridge University Press, 2003.

### Other resources

A complete set of written lecture notes covering the subject will be provided on UTSCanvas.