Edinburgh Mathematics Programme

Level 2/3 Module

MS0105 (FoA) Fundamentals of Analysis (S2)

Introduction

The basic tools of analysis which you will have seen in the FoC (Foundations of Calculus) module are developed in this course by introducing powerful concepts to enable us to solve deep problems of Analysis. One of the most important concepts covered in this module is that of completeness, which arises in many guises. The natural language for discussing this and other important notions, such as convergence, is that of metric spaces. Many results of analysis depend on studying spaces of functions and ways of measuring functions in these spaces. Once again, the language of metric spaces is ideal. This module will introduce these concepts by first understanding how they work in R and R^n, then passing to the metric space setting, with particular emphasis on metric spaces of functions.

Aims

  1. To cover basic ideas of convergence and completeness in R^n.
  2. To discuss the basic notions of topology in R^n.
  3. To introduce metric spaces and the notions of convergence in a metric space and completeness of a metric space.
  4. To introduce the idea of uniform convergence.
  5. To introduce the basic concepts of function spaces as metric spaces, with illustrations.

Prerequisites

MS0095 FoC; concurrent or previous attendance at MS0225 LAG; concurrent or previous attendance at MS0226 MAM would be helpful.

Syllabus summary

Properties of the reals. Cauchy sequences; completeness of R^n. Contraction mapping theorem in R with basic applications. Open and closed subsets of R^n. Definition and examples of metrics. Convergent and Cauchy sequences in metric spaces. The uniform metric on C[a,b] and the notion of uniform convergence. Completeness of C[a,b] with the uniform metric. Examples of other function spaces as metric spaces.

Text

  1. Stoll, M., Introduction to Real Analysis, Addison-Wesley, 1997 (2nd Ed., 2001)

Library shelf reference: QA 300 Sto

Syllabus

  1. Definition of supremum and infimum, basic properties, statement of LUB axiom; consequences: Archimedean property, density of rationals, etc. (2)
  2. The completeness of R. Subsequences, the nested interval and Bolzano-Weierstrass theorems in R. Cauchy sequences, convergent implies Cauchy, Cauchy in R implies convergent. (2.5)
  3. Contraction Mapping Theorem for mappings on closed intervals in R; application to finding approximate roots by iteration. (1)
  4. Sequences in C, convergence equivalent to convergence of real and imaginary parts. The Bolzano-Weierstrass theorem and Cauchy sequences in C. The completeness of C. Inner product and euclidean norm on R^n. Cauchy-Schwarz and triangle inequalities. Convergent and Cauchy sequences in R^n. The completeness of R^n, the Bolzano-Weierstrass in R^n. (2.5)
  5. Infinite series with complex terms. The Cauchy criterion for convergence of infinite series. Absolute convergence implies convergence. Complex power series. Radius and disc of convergence. (2.5)
  6. Definition of open and closed sets in R^n; basics: intersection and union properties, etc., closed if and only if every limit point lies in the set. Continuity for a function f: A --> R with A a subset of R^n, attainment of bounds when A is closed and bounded. (1.5)
  7. Definition of a metric, examples of different metrics on R^n and on C[a,b]. Convergence in metric spaces, Cauchy sequences and completeness. Continuity in metric spaces. Open and closed sets in metric spaces. (3.5)
  8. Pointwise and uniform convergence. Cauchy condition for uniform convergence. Preservation of continuity under uniform convergence (hence C[a,b] complete under uniform metric). Uniform convergence of series of functions, the Weierstrass M-test. (3.5)
  9. The convergence of Fourier series (pointwise/uniform), uniform convergence of Fejer means for continuous functions, convergence in the mean-square metric d_2 of Fourier series of continuous functions and Parseval's Theorem; Weierstrass approximation theorem. (3)

Notes and Links

  1. The principal themes of the module are completeness in its many guises and the ramifications of completeness.
  2. Sections 1 and 2 of the syllabus are meant to be a deeper and fully rigorous treatment of some of the material covered in FoC.
  3. This is the first explicit exposure to complex series, so some care needs to be taken over their treatment.
  4. In this module, metrics provide a language for discussing convergence and completeness, and so the emphasis is on the definition of a metric space and giving plenty of useful examples of metrics. It is not intended that there should be a detailed development of the theory of metric spaces per se.
  5. The examples of metrics to be given should include those corresponding to p=1,2 and infinity, both on euclidean space and on C[a,b]. For p=2 it is useful to make the link with inner product spaces and the Cauchy-Schwarz inequality on inner product spaces. The standard inner product, Cauchy-Schwarz and triangle inequalities on R^n are covered in LAG, but more general inner products are not.
  6. Section 9 introduces the idea that the language of metrics is the appropriate one for concrete problems in analysis such as that of convergence of Fourier series. (The lecturer may wish to tailor the topics suggested here to reflect time remaining and stamina of the class.) In connection with Parseval's theorem and convergence of Fourier series in the d_2 metric, LAG does not currently cover the "closest point to the span of finitely many orthonormal vectors" material. The pointwise convergence of Fourier series may be touched on in MAM (even if not officially in the syllabus for MAM). Suggest to liaise with MAM lecturer on this topic.
  7. An alternative way to round off the course would be to replace the Fourier series material in Section 9 with a discussion of the Contraction Mapping Theorem in metric spaces and give applications of this (e.g. Picard's theorem on the solution of ODEs, the inverse mapping theorem). However, there would be a knock-on effect for AaI if this were done.

Outcomes

  1. Understanding the definitions and concepts of convergence, Cauchy sequences, and completeness in R and R^n.
  2. Understanding the notions of open and closed subsets of R^n and their relationship to convergence.
  3. Familiarity with the notion of a metric and examples of metrics.
  4. Familiarity with the notion of convergence in a metric space and completeness of a metric space.
  5. Ability to test sequences and series of functions for uniform convergence.
  6. Familiarity with the uses of the Weierstrass M-test.
  7. Understanding the notion of function spaces as metric spaces and familiarity with particular examples: C([a,b]) with d_\infty, d_2 and d_1.
  8. Familiarity with the idea that problems of analysis such as convergence of Fourier series can be couched in and resolved using the language of metric spaces.

TAG/AOD

Last revised 03.03.02