MA2006: Real Analysis

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA2006
External Subject Code 100405
Number of Credits 10
Level L5
Language of Delivery English
Module Leader Professor Federica Dragoni
Semester Spring Semester
Academic Year 2022/3

Outline Description of Module

This module follows up on the introduction to mathematical analysis provided in the first-year modules MA1005 Foundations of Mathematics I and MA1006 Foundations of Mathematics II. It develops further important basic concepts of analysis, including convergence of functional sequences and series, the interchangeability (or otherwise) of limits, uniform continuity, as well as their applications to the study of functions defined through series, integrals or differential equations. These concepts form the foundation for later courses on Complex Analysis, Differential Equations and Fourier and Functional Analysis.

On completion of the module a student should be able to

            •           Know firmly all basic definitions, including those of real numbers, supremum and infimum, limit of a sequence, sum of a series, pointwise and uniform convergence of functional sequences and series,continuity, uniform continuity, differentiability, integrability of functions

            •           Explain the logical foundations and mathematical ideas in the development of mathematical analysis, including knowledge of the precise formulations of the main theorems and of their proofs

            •           Apply the theoretical results  to studying the properties of concrete functions, sequences and series

How the module will be delivered

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

  • Weekly face to face classes (e.g. labs, lectures, exercise classes)
  • Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills that will be practised and developed

 

            •           Correct interpretation and manipulation of quantified logical expressions

            •           Ability to understand and assimilate medium-length mathematical arguments

            •           Ability to apply techniques of proofs from a larger toolbox, appreciating the fact that generally                               there is no single approach covering all problems

            •           Ability to recognise the convergence properties of functional sequences and series and the                                 analytical properties of functions defined via limiting processes

            •           Appreciation of the necessity to justify the interchange of limit processes, and ability to provide                           such justification

            •           Ability to apply theoretical results to concrete cases

            •           Use of mathematical language in a logically and rhetorically correct way

 

 

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Spring Semester 90 Real Analysis 2
Class Test 10 Real Analysis N/A

Syllabus content

            •           Basic topology of R^n (open and closed sets)

            •           Completeness of the set of real numbers, the Bolzano-Weierstrass theorem, Cauchy sequences

            •           Weierstrass Extreme Value Theorem and applications to maxima and minima in compact sets

            •           Pointwise and uniform convergence of functional sequences and series, analytical properties of                           limit functions, applications;

            •           Fundamental property and radius of convergence of power series, Abel's limit theorem;

            •           Uniform continuity of functions, integrability, continuity and differentiability of integrals with                                   respect to a parameter

            •           Lipchitz and Holder functions.


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