## 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

### Lecturers

Iain Raeburn, room 515, phone ext 5115, iraeburn@maths.otago.ac.nz

### Office hours

Monday 10am

### Lectures

2 per week in odd weeks, 3 per week in even weeks : Monday at 12pm (all weeks), Wednesday at 12pm (all weeks) and Friday at 9am (even weeks).

### Tutorials

1 hour per week: Thursday 3pm.

### 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 (due on 21 March, 18 April and 23 May) and 50% from a midterm test in class on 2 May.

### 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 endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.