Hilbert Spaces for Scientists and Engineers

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Typical Scheduling: 
Every odd fall semester (starting 2023)

Geometry, convergence, and structure of linear operators in infinite dimensional spaces. Applications to science and engineering, including integral equations and ordinary and partial differential equations.

The three course series MATH 6579, 6580, and 6221 is designed to provide a high level mathematical background for engineers and scientists.

This course is equivalent to MATH 6338. Students should not be able to obtain credit for both MATH 6580 and MATH 6338.

Course Text: 

At the level of Debnath and Mikusi´nski, “Introduction to Hilbert Spaces with Applications, Second Edition."

Topic Outline: 
  • Norms and Banach spaces, L^p and l^p spaces.
  • Bounded operators on normed spaces.
  • Inner products and Hilbert spaces, including orthonormal bases and Fourier series.
  • Operators on Hilbert spaces, adjoints, Riesz Representation Theorem.
  • Compact operators and the Spectral Theorem.
  • Additional topics at instructor’s discretion as time permits. Typical additional topics may include the following (and others):
    • Sturm-Liouville operators.
    • Contraction Mapping Theorem and applications.
    • Fredholm Alternative.
    • Ordinary or partial differential equations.
    • Generalized inverses.
    • Sobolev spaces.