4H: Functional Analysis

Short Description
This is a first course in functional analysis, i.e. the theory of infinite-dimensional vector spaces and of the linear maps between them. In order to be able to discuss the most relevant examples of infinite-dimensional function spaces, the course will begin with a concise development of the Lebesgue integral on the real line.

Assessment
100% Examination.

Reassessment
In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.
Main Assessment In: April/May

Course Aims
This course aims to introduce students to the theory of Banach and Hilbert spaces, that is, infinite-dimensional vector spaces equipped with a norm respectively inner product that turns them into a complete
metric space, and of the operators (i.e. linear maps) between these spaces. These objects arise naturally in applications including wavelets, signal processing and quantum mechanics, and underpin the theory of partial differential equations. The course will also introduce the Lebesgue integral on the real line and the resulting function spaces that provide the most important examples of Hilbert and Banach spaces from the point of view of applications.

Intended Learning Outcomes of Course
By the end of this course students will be able to:
a) Define the concept of a null subset of R and verify that appropriate sets are null;
b) Define the concept of a measurable function and verify that suitable functions are measurable;
c) Define the Lebesgue integral of a real-valued function, compute the integral of appropriate functions;
d) State and prove the major convergence theorems (monotone convergence theorem, Fatou's lemma and the dominated convergence theorem) and use these results to establish integrability of functions and compute suitable integrals;
e) Define the L^p spaces of subsets of R with respect to Lebesgue measure and determine which functions lie in L^p;
f) Define the notion of a norm and use the basic theory of normed linear spaces and operators acting on these spaces to solve simple problems;
g) Determine whether a function on a vector space defines a norm, establish whether such norms are complete and determine when norms are equivalent;
h) Prove the Minkowski inequality and show that the spaces $l^p$ and $L^p$ are complete;
i) Prove that all norms on a finite dimensional vector space are equivalent;
j) State and prove equivalent formulations of boundedness for operators between normed spaces;
k) Define the operator norm, determine whether simple operators between normed spaces are bounded and compute the norm of these operators,prove that the bounded operators between Banach spaces form a Banach space. Solve simple problems involving the operator norm;
l) Define the concepts of linear functionals and dual spaces, use Holder's inequality to exhibit the duality between L^p and L^q (and l^p and l^q) for p^{-1}+q^{-1}=1, show that l_1 is the dual space of c_0 and l_\infty is the dual of l_1;
m) Determine whether a bilinear form on a vector space gives an inner product, develop and use the basic theory of Hilbert spaces and orthogonality, use the Gram-Schmidt process to produce orthonormal sequences and use these in problems, describe and use properties of bases in Hilbert spaces;
n) State and prove the Riesz representation theorem for Hilbert spaces and use this to define and establish properties of the adjoint operator, compute the adjoint of suitable operators;
o) Define what it means for an operator to be compact, show that the compact operators are an ideal and determine whether certain operators are compact. Prove and apply the spectral theorem for compact self-adjoint operators and discuss how this theorem generalises results for finite dimensional matrices.