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Modern mathematics and physics rely on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called ``analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable.
To introduce students to the fundamentals of mathematical analysis and to reading and writing mathematical proofs. At the end of this course, students should: understand the axiomatic foundation of the real number system, in particular the notion of completeness and some of its consequences; understand the concepts of limits, continuity, compactness, differentiability, and integrability, rigorously defined; be able to use results and techniques involving these concepts to solve a variety of problems, including types of problems that they have not seen previously; know how completeness, continuity, and other notions are generalised from the real line to metric spaces; and appreciate the Contraction Principle in abstract metric space theory as a powerful tool to solve concrete problems, especially in differential equations. Students should also have attained a basic level of competency in developing their own mathematical arguments and communicating them to others in writing.
Topics covered are: Basic set theory. The real numbers, least upper bounds, completeness and its consequences. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions, uniform continuity. Differentiation, the Mean Value Theorem. Sequences and series of functions, pointwise and uniform convergence. Power series and Taylor series. Metric spaces: basic notions generalised from the setting of the real numbers. The space of continuous functions on a compact interval. The Contraction Principle. Picard's Theorem on the existence and uniqueness of solutions of ordinary differential equations.
This course is essential for students wishing to study more advanced mathematical analysis, the theory of differential
equations, differential geometry, topology, or mathematical physics.
The course also provides widely applicable training in constructing and writing rigorous arguments.