Mathematics

Functional Analysis (L.6)

Module code: G1029
Level 6
15 credits in autumn semester
Teaching method: Lecture
Assessment modes: Unseen examination, Coursework

This module serves as an introduction to the abstract area of Functional Analysis. You will study the theory of Banach and Hilbert spaces, fundamental results concerning linear functions on these spaces such as the open mapping and closed graph theorems, the uniform boundedness principle, and the Hahn-Banach theorem.

Functional Analysis studies vector spaces endowed with certain special structures (normed and inner product spaces), as well as the linear functions between them. These structures generalise those found in Euclidean spaces but, unlike Euclidean spaces and their linear functions studied in Linear Algebra, the spaces considered here need not be finite-dimensional.

These infinite-dimensional vector spaces include many examples that are important for applications, and their study gives rise to the theory of Functional Analysis which forms the theoretical foundation for most of the mathematical analysis and theory of partial differential equations underpinning the mathematical treatment of models in applied science.

Module learning outcomes

• A successful student should:know the basic facts and the definitions about Hilbert and Banach spaces and their duals;
• be able to state and sketch the ideas of the proofs of the following basic theorems and principles: Baire, Banach-Steinhaus, Hahn- Banach, closed graph, open mapping; contraction mapping theorem.