Content: This covers three topics:

• Integration
• Convergence of sequences and series of functions
• Norms

The idea behind integration is to compute the area under a curve. The fundamental theorem of calculus gives the precise relation between integration and differentiation. However, integration involves taking a limit, and the deeper properties of integration require a precise and careful analysis of this limiting process. This module proves that every continuous function can be integrated, and proves the fundamental theorem of calculus. It also discusses how integration can be applied to define some of the basic functions of analysis and to establish their fundamental properties.

Many functions can be written as limits of sequences of simpler functions (or as sums of series): thus a power series is a limit of polynomials, and a Fourier series is the sum of a trigonometric series with coefficients given by certain integrals. The second part of the module develops methods for deciding when a function defined as the limit of a sequence of other functions is continuous, differentiable, integrable, and for differentiating and integrating this limit. Norms are used at several stages and finally applied to show that a Differential Equation has a solution.

Aims:

• To develop a good working knowledge of the construction of the integral of regulated functions
• To study the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions
• To use the concept of norm in a vector space to discuss convergence and continuity there

Objectives: By the end of the course the student should understand:

• The need for a rigorous theory of integration, and that this can be developed for regulated functions by approximating the area under the graph by rectangles
• Uniform and pointwise convergence of functions together with properties of the limit function
• Be able to prove the main results of integration: any continuous function can be integrated on a bounded interval and the Fundamental Theorem of Calculus
• Prove and apply the Contraction Mapping Theorem

Books:
There is no book that covers the whole module Analysis III in the style of the lectures. However, it may be useful to consult titles suggested on the module page for Regulated Functions, Uniform Convergence or the Fundamental Theorem of Calculus. Students registered for this module may access the relevant chapters of books scanned under copyright.