Part A Topology, Part B Banach Spaces and Hilbert Spaces.
This course builds on B4.1 and B4.2, by extending the theory of Banach spaces and operators. As well as developing general methods that are useful in operator theory, we shall look in more detail at the structure and special properties of "classical'' sequence spaces and function spaces.
Dr David Seifert
Normed vector spaces and Banach spaces. Dual spaces. Direct sums and complemented subspaces. Quotient spaces and quotient operators.
Baire's Category Theorem and its consequences (review).
Hahn-Banach extension and separation theorems. The bidual space. Reflexivity. Completion of a normed vector space.
Convexity and smoothness of norms. Classical Banach spaces and their duals.
Weak and weak* topologies. The Banach-Alaoglu theorem. Goldstine's theorem. Weak compactness. The Schur property of \(\ell^1\).
Compactness in normed vector spaces. Compact operators. The Arzelà-Ascoli theorem. Schauder's theorem on compactness of dual operators. Completely continuous operators.
The Closed Range Theorem. Fredholm theory: Fredholm operators; the Fredholm index; perturbation results; the Fredholm Alternative. Spectral theory of compact operators. The spectral theorem for compact self-adjoint operators.
Schauder bases; examples in classical Banach spaces.