Hilbert spaces
4 points * Second semester * Clayton * Prerequisites: MAP2011, MAT2020; MAP3071 is also recommended
Objectives On the completion of this subject students will be able to understand the basic concepts of complex vector spaces, normed and inner product spaces as well as Banach and Hilbert spaces; use the concept of orthogonality in an inner product or Hilbert space, in particular recognise orthonormal systems and apply the Gram-Schmidt orthonormalisation process; master some basic properties of complete orthonormal systems and Fourier expansions and their relation to classical (trigonometric) Fourier expansions; apply the above theoretical results to spaces of continuous functions on a closed interval, with different norms and inner products; understand the concepts and basic properties of bounded linear operators and functionals, including the Riesz representation theorem and the adjoint of an operator; know elements of spectral theory, in particular for self-adjoint operators; reproduce proofs of theorems concerning Hilbert spaces and linear operators in some straightforward cases.
Synopsis Inner product spaces, orthogonality. Hilbert spaces. Orthonormal systems, Gram-Schmidt process, Fourier expansions. Bounded linear operators and functionals. Riesz representation theorem. Spectral theory for self-adjoint operators: applications.
Assessment Examinations (1.5 hours): 70% * Assignments: 30%
Prescribed texts
Young N An introduction to Hilbert spaces CUP, 1988
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