Aims & Learning Objectives:
Aims: To introduce the theory of infinite-dimensional normed vector spaces, the linear mappings between them, and spectral theory.
By the end of the unit, the students should be able to state and prove the principal theorems relating to Banach spaces, bounded linear operators, compact linear operators, and spectral theory of compact self-adjoint linear operators, and apply these notions and theorems to simple examples.
Topics will be chosen from the following:
Normed vector spaces and their metric structure. Banach spaces. Young, Minkowski and Holder inequalities. Examples - IRn, C[0,1], lp, Hilbert spaces. Riesz Lemma and finite-dimensional subspaces. The space B(X,Y) of bounded linear operators is a Banach space when Y is complete. Dual spaces and second duals. Uniform Boundedness Theorem. Open Mapping Theorem. Closed Graph Theorem. Projections onto closed subspaces. Invertible operators form an open set. Power series expansion for (I-T)-1. Compact operators on Banach spaces. Spectrum of an operator - compactness of spectrum. Operators on Hilbert space and their adjoints. Spectral theory of self-adjoint compact operators. Zorn's Lemma. Hahn-Banach Theorem. Canonical embedding of X in X*
* is isometric, reflexivity. Simple applications to weak topologies.