Introduction to Hilbert Spaces

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Typical Scheduling: 
Every summer and fall semesters

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.

Course Text: 

At the level of Naylor and Sell, Linear Operator Theory in Engineering and Science or Debnath and Mikusinski, Introduction to Hilbert Spaces with Applications

Topic Outline: 
  • Background: Vector spaces, dot products, norms, Cauchy-Schwartz inequality
  • Contrast the geometry of R^n, R^\infty, l^2, L^2(IR), and other spaces
  • Complete orthonormal sequences, Fourier series, Bessel's and Parseval's inequality
  • Projections: closest point projections, linear projections, non-expansive projections, orthogonal projections, and self-adjoint projections
  • Bounded linear functions, Riesz representation theorem, and the Lax-Milgram theorem
  • Characterizations of finite dimensional and of self-adjoint, normal, compact, or closed linear operators.
  • A structure for unbounded linear operators, Sturm-Liouville operators
  • Contraction Mapping Theorem and applications
  • Various topics depending on the interest of the instructor: Fredholm Alternative Theorems, control problems, ordinary or partial differential equations, semigroups of operators, generalized inverses, reproducing kernel Hilbert spaces, etc.
  • Normed and Sobolev spaces