# Hilbert Spaces

University of Otago

## Course Description

• ### Course Name

Hilbert Spaces

• ### Host University

University of Otago

• ### Location

Dunedin, New Zealand

Mathematics

• ### Language Level

Taught In English

• ### Prerequisites

MATH 201 and MATH 202

• ### Course Level Recommendations

Upper

ISA offers course level recommendations in an effort to facilitate the determination of course levels by credential evaluators.We advice each institution to have their own credentials evaluator make the final decision regrading course levels.

### Hours & Credits

• Credit Points

18
• Recommended U.S. Semester Credits
3 - 4
• Recommended U.S. Quarter Units
4 - 6
• ### Overview

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 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.

Course 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

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.

### Course Disclaimer

Courses and course hours of instruction are subject to change.

Eligibility for courses may be subject to a placement exam and/or pre-requisites.

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Credits earned vary according to the policies of the students' home institutions. According to ISA policy and possible visa requirements, students must maintain full-time enrollment status, as determined by their home institutions, for the duration of the program.

Please reference fall and spring course lists as not all courses are taught during both semesters.

Availability of courses is based on enrollment numbers. All students should seek pre-approval for alternate courses in the event of last minute class cancellations

Please note that some courses with locals have recommended prerequisite courses. It is the student's responsibility to consult any recommended prerequisites prior to enrolling in their course.