620-312 Linear Analysis

620-311.

The most important topic of this subject is integration. Students meet this concept in a calculus course where an integral is defined as a Riemann integral. Although a Riemann integral is useful in many areas of mathematics, it is not adequate for many problems of modern analysis. The aim of the subject is to introduce students to the Lesbesgue theory of integration and measure theory. Included in this subject is an introduction to the fundamental concepts of functional analysis. Functional analysis is the common name for the study of infinite dimensional vector spaces and the linear maps between them. What distinguishes this subject from linear algebra is the role of topological considerations. These topics are not only beautiful and interesting but are also useful in other branches of mathematics such as probability theory, partial differential equations and quantum mechanics.

Topics include construction of measures, measurable functions, Lesbesgue integrals, convergence theorems, L^p-spaces, Fubini's theorem, normed spaces and Banach spaces, inner product and Hilbert spaces, linear functionals and linear operators.

Up to 36 pages of written assignments due during the semester (either 0% or 20%); a 3-hour written examination in the examination period (80% or 100%). The relative weighting of the examination and the assignments will be chosen so as to maximise the student's final mark.