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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
Paper code MATH301
Subject Mathematics
EFTS 0.1500
Points 18 points
Teaching period First Semester
Domestic Tuition Fees (NZD) $851.85
International Tuition Fees (NZD) $3,585.00

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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
Professor Iain Raeburn
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
  • The Riesz representation theorem and adjoints
  • Properties of adjoints: unitaries, isometries and projections
Textbooks
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.

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Timetable

First Semester

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

Lecture

Stream Days Times Weeks
Attend
L1 Monday 12:00-12:50 9-15, 17-22
Wednesday 12:00-12:50 9-15, 17-22
Friday 09:00-09:50 10, 12, 14, 18, 20, 22

Tutorial

Stream Days Times Weeks
Attend
T1 Thursday 15:00-15:50 10-15, 17-22

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
Paper code MATH301
Subject Mathematics
EFTS 0.1500
Points 18 points
Teaching period First Semester
Domestic Tuition Fees Tuition Fees for 2018 have not yet been set
International Tuition Fees Tuition Fees for international students are elsewhere on this website.

^ Top of page

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
Professor Iain Raeburn
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
Textbooks
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.

^ Top of page

Timetable

First Semester

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

Lecture

Stream Days Times Weeks
Attend
L1 Monday 12:00-12:50 9-13, 15-22
Wednesday 12:00-12:50 9-13, 15-16, 18-22
Friday 09:00-09:50 10, 12, 16, 18, 20, 22

Tutorial

Stream Days Times Weeks
Attend
T1 Thursday 15:00-15:50 10-13, 15-22