MAU23203 - Analysis in Several Real Variables Dr. David R. Wilkins Topics in Real Analysis

Topics in Real Analysis

These accounts of various topics in real analysis extend the module content of module MAU23203, providing more detailed discussion of topics in real analysis related to those included in the MAU23203 course notes.

Those enrolled in MAU23203 in the academic year 2021/22 are not expected to master the material included in these accounts, unless the material is explicitly encluded in the designated module content for the academic year 2021/22.

Ordered Fields and the Real Number System

This account of the theory of ordered fields was included in the module content for module MA2321 in Michaelmas Term 2021. It provides a more detailed account of the theory of ordered fields than is found in the course notes for MAU23203 in 2021/22. In particular, the account available here includes a reasoned argument intended to demonstrate that if an ordered field containing the field of rational number satisfies the Axiom of Archimedes, and if every decimal expansion determines an element of that ordered field, then the ordered field must necessarily satisfy the Least Upper Bound Principle.

Dedekind Sections and the Real Number System

Available here is a somewhat lengthy account of the construction of the real number system using Dedekind Sections (also known as Dedekind Cuts). The account aims to show that, provided operations of addition and multiplication are defined in an appropriate fashion on the set of Dedekind Sections, and if the ordering of Dedekind Sections is defined in the most natural fashion, then the set of Dedekind Sections, with these operations of addition and multiplication, and with this ordering, satisfies all of the axioms that a Dedekind-complete ordered field is required to satisfy.

Basic Properties of the Riemann Integral

This account of the theory of the Riemann integral, in the approach pioneered by Darboux, begins from first principles, and concludes with a proof that bounded increasing functions are Riemann-integrable on bounded intervals. The account does not include a proof of the result that continuous functions are Riemann-integrable, and it does not include any statement or proof of the Fundamental Theorem of Calculus.

Convergence and Continuity in the Theory of Metric Spaces

This account of convergence and continuity in the theory of metric spaces follows closely the relevant portions of the course notes for module MAU23203 (Analysis in Several Real Variables) but systematically modifies the statements and proofs of many of the results concerning convergence and continuity so as to present them in the more general context provided by the theory of metric spaces.

The Contraction Mapping Theorem

It is shown that every contraction mapping defined on a complete metric space has a unique fixed point.

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Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.