General Prerequisites: Students wishing to take this course are expected to have a thorough understanding of the basic theory of normed vector spaces (including properties and standard examples of Banach and Hilbert spaces, dual spaces, and the Hahn-Banach theorem) and of bounded linear operators (ideally including the Open Mapping Theorem, the Inverse Mapping Theorem and the Closed Graph Theorem). Some fluency with topological notions such as (sequential) compactness and bases of topological spaces will also be assumed, as will be basic familiarity with the Lebesgue integral. A number of these
prerequisites will be reviewed (briefly) during the course, and there will be a document available on the course webpage summarising most of the relevant background material.
Course Overview: This course builds on what is covered in introductory courses on Functional Analysis, 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.
Learning Outcomes: By the end of this course, students will be able to:
1. Establish and use both extension and separation versions of the Hahn Banach Theorem, and geometric properties of the norm, to obtain dualities between embeddings and quotients and give characterisations of reflexivity.
2. Work with the weak and weak*-topologies on Banach spaces, establish and use the Banach-Alaoglu theorem, relating this to characterisations of reflexivity, and describe closures in both norm and weaker topologies using annihilators, and preannhilators.
3. Manipulate properties of compact and Fredholm operators on Banach and Hilbert spaces, to establish and use the Fredholm alternative, and obtain spectral theorems for compact operators both in abstract and concrete settings.
Course Synopsis: Normed vector spaces and Banach spaces. Dual spaces. Direct sums and complemented subspaces. Quotient spaces and quotient operators.
The Baire Category Theorem and its consequences (review).
Hahn-Banach extension and separation theorems. The bidual space. Reflexivity. Completion of a normed vector space.
Weak and weak* topologies. The Banach-Alaoglu theorem. Goldstine's theorem. Equivalence of reflexivity and weak compactness of the closed unit ball. The Schur property of \(\ell^1\) Weakly compact operators.
Compactness in normed vector spaces. Compact operators. 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.
