Analysis 2

Abstract

<p><b>Part 2 consists of three units:</b></p>

<p>Unit 4 – Real Analysis </p>

<p>In this unit we define and give examples of general metric spaces which show that they occur abundantly in mathematics. We then look at the structure of a general metric space a long the lines of unit 1. In addition we introduce the concept of compactness and its effects on continuity of functions.</p>

<p>Unit 5 – Topology</p>

<p>The structures of topological spaces are studied along side those of metric spaces. However the main difference here is that the axioms that define a metric space are dependent on the concept of distance whereas in the axioms that define a topological space the concept of distance is absent. In particular the study of the twin concepts of convergence and continuity brings out this difference very well. Finally a look at different topologies like product or quotient topology endowed on a set is essential in this unit.</p>

<p>Unit 6/7 – Measure Theory</p>

<p>In this unit we start with the study of both the Lebesgue outer measure and the real line before we look at the Lebesgue measurable subsets of the real line. A look at the sigma algebra of subsets of a given underlying set gives rise to measurable space on which we can also study a class of functions called measurable functions. A part from the Lebesgue measure we also study an abstract measure leading to an abstract measure space on which we introduce an abstract integral. Finally a brief comparison of the Lebesgue integral and the well known Riemann integral is also essential in this unit.</p> <p> <b>In addition to downloading the files below, you can also view them directly:</b> </p> <ul> <li><a href="http://www.scribd.com/doc/44852727/Analysis-2" target="_blank">Module part 2 (Scribd)</a></li> <li><a href="http://www.youtube.com/watch?v=Ac_MRT44fx0" target="_blank">Intro to module (YouTube)</a></li> <li><a href="http://www.youtube.com/watch?v=dfI7APUjptE" target="_blank">Overview of module (YouTube)</a></li> </ul>