- Lecturer: Jan Kristensen
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 term: Michaelmas
Course lecture information: 16 lectures
Course weight: 1
Course level: M
Assessment type: Written Examination
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. Work with series in Banach spaces and use Schauder bases when they exist to establish reflexivity and to distinguish separable Banach spaces.
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. Work with series in Banach spaces and use Schauder bases when they exist to establish reflexivity and to distinguish separable Banach spaces.
Course synopsis:
Hamel bases and existence of unbounded linear functionals on an infinite-dimensional Banach space. Direct sums and complemented spaces. Proof that c0 is not (topologically) complemented in ℓ∞. Quotient spaces, quotient operators and embeddings. The Successive Approximations Lemma and the Closed Range Theorem.
Hahn-Banach extension theorem for normed spaces over C. Hahn-Banach separation theorems. Extreme points for convex sets and the Krein-Milman Theorem.
The bidual space. Reflexivity. Completion of a normed space. Uniform convexity and smoothness of norms. Radon-Riesz property. The Milman-Pettis Theorem. Examples: The Clarkson and Beurling-Hanner inequalities (without proofs).
Weak and weak* topologies. The Banach-Alaoglu Compactness Theorem. The Goldstine Theorem. The Kakutani Theorem. The Eberlein-Smulian Theorem (statement only). Comparison of weak and norm convergence for sequences and the Schur property for ℓ¹.
Series in Banach spaces: absolute, unconditional and conditional convergence. The DvoretzkyRogers Theorem (statement only). Orlicz’s Theorem about unconditional convergence in 𝐿ᵖ (statement only). Schauder bases and examples in classical Banach spaces. Reflexivity in terms of Schauder bases. Riesz bases and examples of generalized Fourier series.
Hahn-Banach extension theorem for normed spaces over C. Hahn-Banach separation theorems. Extreme points for convex sets and the Krein-Milman Theorem.
The bidual space. Reflexivity. Completion of a normed space. Uniform convexity and smoothness of norms. Radon-Riesz property. The Milman-Pettis Theorem. Examples: The Clarkson and Beurling-Hanner inequalities (without proofs).
Weak and weak* topologies. The Banach-Alaoglu Compactness Theorem. The Goldstine Theorem. The Kakutani Theorem. The Eberlein-Smulian Theorem (statement only). Comparison of weak and norm convergence for sequences and the Schur property for ℓ¹.
Series in Banach spaces: absolute, unconditional and conditional convergence. The DvoretzkyRogers Theorem (statement only). Orlicz’s Theorem about unconditional convergence in 𝐿ᵖ (statement only). Schauder bases and examples in classical Banach spaces. Reflexivity in terms of Schauder bases. Riesz bases and examples of generalized Fourier series.