Department:Mathematics and Statistics
Level:Part II (any yr)
Course Convenor:Professor JM Lindsay
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Prior to MATH317, the student must have successfully completed:
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CMod descriptionback to top
Inner products on Cn. Orthogonal vectors in Cn. Linear
maps and matrices. The adjoint of a matrix. Unitary and self-adjoint matrices
in Cn. Spectral theorem for self-adjoint matrices on Cn (statement).
Functions on Rn. The derivative and Hessian matrix. Quadratic forms. Normed
vector spaces. Inner products. Convergence of sequences in normed vector
spaces. Completeness. Hilbert space. Completeness of Hilbert sequence
space. Cauchy-Schwarz Inequality. Orthogonality. Parallelogram law. Sums of
orthogonal series in Hilbert space. Gram-Schmidt process. Examples.
Curriculum Design: Outline Syllabusback to top
- Normed linear spaces: definition and examples. Sequences and series. Closest points. Convex sets. Continuity and norms of linear mappings. The closure of a set.
- Inner products. The Cauchy-Schwarz inequality and the derived norm. Examples. Linear mapping on inner product spaces.
- Orthogonality. Finding the closest point in a linear subspace. Orthonormal sets. The Gram-Schmidt process. Bessel's inequality. Fourier series.
- Completeness. Theorem on closest points in a closed, convex subset. Orthogonal complements. Representation of linear functionals. Isometry of all separable Hilbert spaces.
- The adjoint of a linear operator. Kernel and range. Quadratic forms. The spectral theorem in finite dimensions.
Curriculum Design: Pre-requisites/Co-requisites/Exclusionsback to top
Prerequisites: MATH220; MATH210 or MATH211
Educational Aims: Subject Specific: Knowledge, Understanding and Skillsback to top
This course introduces the student to an area of Mathematics in which the concepts of linear algebra, analysis and geometry are harnessed together. It is shown how this leads to powerful and elegant generalizations of earlier results, many of which are fundamental to modern applications of analysis.
Curriculum Design: Select Bibliographyback to top
N J Young, An Introduction to Hilbert Space, Cambridge University Press, 1988.