# MATH301 Hilbert Spaces

An introduction to Hilbert spaces and linear operators on Hilbert spaces, grounded in applications to Fourier analysis, spectral theory and operator theory.

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 paper will introduce students to the basic techniques of functional analysis in the context of Hilbert spaces and linear operators on Hilbert spaces. The paper 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.

Paper title Hilbert Spaces MATH301 Mathematics 0.15 18 points Semester 1 (On campus) \$929.55 Tuition Fees for international students are elsewhere on this website.
Prerequisite
MATH 201 and MATH 202
Schedule C
Arts and Music, Science
Eligibility
This paper is particularly relevant to Mathematics and Physics majors.
Contact
maths@otago.ac.nz
Teaching staff

Paper Structure

Main 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
• Fixed point theorems and applications
• Closed subspaces and orthogonal projections, convexity and least squares approximation
• Orthonormal bases and the reconstruction formula
• The Fourier basis and Fourier series
• Diagonalisation of compact self-adjoint operators
Textbooks

Textbooks are not required for this paper.

Course outline
View course outline for MATH 301
Critical thinking.
Learning Outcomes

Students who successfully complete this paper will

• Understand the basic techniques of functional analysis in the context of Hilbert spaces and linear operators on Hilbert spaces
• Gain experience in modern mathematical reasoning and writing

## Timetable

### Semester 1

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
Other

#### Lecture

Stream Days Times Weeks
Attend
A1 Monday 12:00-12:50 9-15, 18-22
Wednesday 12:00-12:50 9-15, 17-22
Friday 09:00-09:50 9, 11, 14, 17-18, 20, 22

#### Tutorial

Stream Days Times Weeks
Attend
A1 Thursday 15:00-15:50 10-15, 17-22

An introduction to Banach and Hilbert spaces, linear operators and their applications.

MATH 301 extends the techniques of linear algebra and real analysis to study problems of an intrinsically infinite-dimensional nature. 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 paper will introduce students to the basic techniques of functional analysis. The paper will be grounded in applications that will reinforce the students' understanding of linear algebra and real analysis; and will give them training in modern mathematical reasoning and writing.

Paper title Introduction to Functional Analysis MATH301 Mathematics 0.15 18 points Semester 1 (On campus) Tuition Fees for 2023 have not yet been set Tuition Fees for international students are elsewhere on this website.
Prerequisite
MATH 201 and MATH 202
Schedule C
Arts and Music, Science
Eligibility
This paper is particularly relevant to Mathematics and Physics majors.
Contact

mathsstats@otago.ac.nz

Teaching staff

Textbooks

Textbooks are not required for this paper.

Critical thinking.
Learning Outcomes

Students who successfully complete this paper will:

• Understand the basic techniques of functional analysis
• Gain experience in modern mathematical reasoning and writing

## Timetable

### Semester 1

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
Other

#### Lecture

Stream Days Times Weeks
Attend
A1 Monday 12:00-12:50 9-14, 16-22
Wednesday 12:00-12:50 9-14, 16-22
Friday 09:00-09:50 9, 11, 13, 17-18, 20, 22

#### Tutorial

Stream Days Times Weeks
Attend
A1 Thursday 15:00-15:50 10-14, 16-22