This module continues the development of rigorous integration theory and provides an elementary introduction to Hilbert spaces. It provides a sound foundation in analysis for students wishing to continue their mathematical studies at graduate level.
Hilbert spaces. Inner products, the Cauchy-Schwartz inequality, orthogonality. Fourier series with respect to an orthonormal basis, applications to solving differential equations. Parseval's theorem and its application to the Fourier transform.
Orthogonal complements and projections; best approximations and its applications in numerical analysis. Riesz's representation theorem, braket notation.
Bounded linear operators on a Hilbert space. Introduction to Spectral theory: eigenvalues and eigenvectors, spectrum, spectral radius. Neumann series, the spectral radius as a "measure" of an operator, applications to convergence of iterative algorithms. Mathematical formulation of quantum mechanics.
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Formal Examination | 100 |