Real Analysis - 2024 entry

Infinite processes appear naturally in many contexts, from science and engineering to economics. From solving the equation that finds the wave function of a quantum system in physics, processing sensor data in engineering, to calculating prices for options in economics, at the foundation of all of these are infinite processes and the pure mathematics developed to rigorously and correctly handle these processes. That field of pure mathematics is called analysis, and the central object of study in analysis is the limit which further extends to the notions of convergence, continuity, differentiation, and integrability.  

In this module, you will be introduced to the pioneering work of Cauchy, Riemann and many other notable mathematicians. By building on material from the first year, we will carefully and rigorously develop notions first in the context of real variables. In particular, we will develop how to rigorously handle real-variable differentiation, Riemann integration, power series, and basic notions of point set topology.

 
The material in this module is a prerequisite for the study of Complex Analysis (MTH2009), Topology and Metric Spaces (MTH3040), Integral Equations (MTH3042), Fractal Geometry (MTHM004), Functional Analysis (MTH3050). It is recommended for those studying Dynamical Systems and Chaos (MTHM018), and is the basis for applications in economics, science, and engineering. 

Pre-requisite modules: MTH1001; MTH1002 (or equivalent)
 

Analysis is the theory that underpins all continuous mathematics. The objective of this module is to provide you with a logically based introduction to real analysis. The primary objective is to define all the basic concepts clearly and to develop them sufficiently to provide proofs of useful theorems. This enables you to see the reason for studying analysis, and develops the subject to a stage where you can use it in a wide range of applications.
 

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 state and prove key theorems in real analysis using a rigorous approach;
2 develop proofs related to topological concepts such as limits and connectedness;
3 understand the basis of integration of functions of a real variable.

Discipline Specific Skills and Knowledge:
4 apply fundamental mathematical concepts, manipulations and results in analysis;
5 formulate rigorous arguments as part of your mathematical development;

Personal and Key Transferable/ Employment Skills and Knowledge:
6 think analytically and use logical argument and deduction;
7 communicate your ideas effectively in writing and verbally;
8 manage your time and resources effectively.

 

- Topology on R; Bolzano-Weierstrass theorem

- Epsilon-delta function limits; continuity; differentiability in R

- Function classes: C^k, C^infinity etc; Lipschitz continuity

- Review of epsilon-N sequence limits, Cauchy sequences; series of real numbers, sequences and series of functions;

- Formal theory of Riemann integration; integrability of monotonic functions and continuous functions; problems interchanging limits in general

- Continuity and differentiability in R^n, inverse and implicit function theorems.
 

Deferrals: Reassessment will be by coursework and/or exam in the deferred element only.  For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only.  As it is a referral, the mark will be capped at 40%   

Web based and Electronic Resources:

ELE: http://vle.exeter.ac.uk

William F. Trench, Introduction to Real Analysis, freely downloadable here: https://digitalcommons.trinity.edu/mono/7/

 

Reading list for this module: