Hilbert Spaces

University of Otago

Course Description

  • Course Name

    Hilbert Spaces

  • Host University

    University of Otago

  • Location

    Dunedin, New Zealand

  • Area of Study

    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.

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