One half of this course examines integration in more detail, and develops the theory of measurability, measure and the Lebesgue integral on Euclidean spaces. The concepts of measure and integral are then extended to the more abstract context of measure spaces, and the monotone and dominated convergence theorems for the Lebesgue integral are proved. The other half introduces Hilbert space, Banach spaces, dual spaces, and linear operators, and explores the interaction between linear algebra and analysis in the study of infinite dimensional spaces.
Learning Outcomes
By the end of the module the student should be able to:
Understand the limitations of Riemann integration and how the Lebesgue integral overcomes many of these
Manipulate abstract sets to prove results on measurability and measure
Apply the main theorems on Lebesgue integration to solve problems of convergence of functions
Work in infinite dimensional Hillbert and Banach spaces, finding bases of Hilbert space, and using different types of convergence
Combine ideas from algebra and analysis to solve problems in functional analysis
Level M students will explore the subject beyond the taught syllabus.
Assessment
22791-01 : Raw Module Mark : Coursework (100%)
Assessment Methods & Exceptions
Assessments: 22791-01: Exam : Exam (CT) - Written Unseen (90%) 22791-02 : Continuous Assessment : Coursework or class Tests (10%)