Lecturer: Professor T T West
Date: 1996-97
Groups: Optional JS and SS Mathematics, SS Two-subject Moderatorship
Prerequisites: 211, 212, 221
Duration: 21 weeks
Lectures per week: 3
Assessment: Final Exam 70%/Project 30% OR Final Exam 100%
Examinations: One 3-hour examination
Introduction: Linear algebra, matrices and linear operators on finite dimensional vector spaces, eigenvalue theory.
Metric Spaces: Axioms, examples, inequalities completeness, separability.
Linear Spaces: Axioms, norms, examples, Banach spaces, subspaces, quotient-spaces, direct sums.
Linear Functionals, Dual Spaces: Boundedness and continuity of linear functionals, dual of l_p, Hahn-Banach extension theorem for linear functionals defined on a subspace.
Hilbert Spaces: Inner-products, orthogonality, complete orthonormal systems, Fourier coefficients, equivalence of separable Hilbert spaces.
Contraction Mappings: Fixed point theorems, examples.
Linear Operators: Definition, continuity and norm, restriction and quotient operators, adjoint operators, projections.
Vector Valued Analytic Functions: Vector valued integration, Cauchy's theorem, contour integrals.
Spectral Theory: Spectrum, resolvent spectral radius, applications of vector valued analytic function theory, operational calculus, spectral theorem.
Banach Algebras: Definition, examples, Gelfand representation, spectral theorem for normal operators.