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|
|Teaching period||First Semester|
|Domestic Tuition Fees (NZD)||$904.05|
|International Tuition Fees (NZD)||$3,954.75|
- MATH 201 and MATH 202
- Schedule C
- Arts and Music, Science
- This paper is particularly relevant to Mathematics and Physics majors.
- More information link
- View more information about MATH 301
- Teaching staff
To be advised
- 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
- Closed subspaces and orthogonal projections, convexity and least squares approximation
- Orthonormal bases and the reconstruction formula
- The Fourier basis and Fourier series
- Uniform convergence and the Fourier series of smooth functions
- Diagonalisation of compact self-adjoint operators
- Text books are not required for this paper.
- Course outline
- View course outline for MATH 301
- Graduate Attributes Emphasised
- Critical thinking.
View more information about Otago's graduate attributes.
- Learning Outcomes
- To understand the basic techniques of functional analysis in the context of Hilbert spaces and linear operators on Hilbert spaces
- To gain experience in modern mathematical reasoning and writing.