## MATH301 Hilbert Spaces

First Semester |

### Paper details

MATH 301 extends the techniques of linear algebra and real analysis to study problems of an intrinsically infinite-dimensional nature. A Hilbert space is a vector space with an inner product that allows length and angles to be measured; the space is required to be complete (in the sense that Cauchy sequences have limits) so that the techniques of analysis can be applied. Hilbert spaces arise frequently in mathematics, physics, and engineering, often as infinite-dimensional function spaces. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (with applications to signal processing and heat transfer) and many areas of pure mathematics including topology, operator algebra and even number theory.

The course will introduce students to the basic techniques of functional analysis in the context of Hilbert spaces and linear operators on Hilbert spaces. The course will be grounded in applications to Fourier analysis, spectral theory and operator theory, will reinforce the students' understanding of linear algebra and real analysis, and will give them training in modern mathematical reasoning and writing.

### Potential students

This paper is particularly relevant to Mathematics and Physics majors.

### Prerequisites

MATH 201 (Real Analysis) and MATH 202 (Linear Algebra).

### Course Outline

The paper will cover the following topics:

- Inner-product spaces, the Cauchy Schwarz inequality and the norm
- Cauchy sequences and completeness, examples of Hilbert spaces
- Normed spaces and bounded linear operators
- Closed subspaces and orthogonal projections, convexity and least squares approximation
- Orthonormal bases and the reconstruction formula
- The Fourier basis and Fourier series
- Compact self-adjoint operators

### Lecturer

Dr Timothy Candy, room 216, phone ext 7781, tcandy@maths.otago.ac.nz

### Lectures

Monday at 12pm (all weeks), Wednesday at 12pm (all weeks) and Friday at 9am (on weeks 9, 11, 13, 18, 20, 22) in room MA241.

### Tutorials

1 hour per week: Thursdays 3pm in room MA241.

### Exercises

Exercise sets will be posted chapter by chapter as we cover the relevant material in class. If you get stuck on a question, ask about it in the tutorial first, then seek further help if needed.

### Internal assessment

Internal assessment is made up of 50% from 3 assignments and 50% from a midterm test in class.

### Assignments and writing mathematics

In the assignments (and the test and exam!), marks will be awarded for correct working, logical setting out, appropriate explanations and presentation — and not just the final answer. The aim of this is to develop your technical-writing skills and for you to learn to present mathematics in a professional way. So pay attention to neatness, grammar, clarity of argument, use of notation and so forth. You should be writing in sentences and paragraphs, just like we do in class.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, 7E/10 + (3A + 3T)/20)**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Tests mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.