Time : Tue 8,9
Credits : 2
Textbook :
1. W. A. Sutherland, Introduction to metric and topological spaces
2. Yuri Safarov, Lecture Notes of CCM321 REAL ANALYSIS II (Download here)
The current version of Calculus courses (MATH4006-4009) offered for students from external departments emphasises on the computational proficiency and lack discussion on the rigorous treatment of the theory of calculus. However, such rigorous treatment is getting increasingly interested by a board range of students from various departments ; for instance, students from physics department work intensively over Hilbert spaces or students from the economics department need to learn various notions of `point-set topology’ to pursue higher level courses in their department.
Our current “Introduction to Analysis” course is primarily designed for students of the math department and hence apparently inaccessible for many of these students who have only completed MATH4006-9. This course aims to bridge the gap a student may have in transition from calculus courses to analysis courses : introducing basic notions of analysis and, most importantly, getting them a taste of how rigorous argument in mathematics is done and hopefully smoothen their transition to advanced courses from our department.
The course begins by introducing the concept of a metric space—a set equipped with a well-defined way to measure the “distance” between its elements. While the theory will be developed abstractly, two key examples will illustrate these ideas: R^n with the Euclidean metric and function spaces with the supremum metric.
We will explore the convergence of sequences in metric spaces, define open and closed sets, and explain continuity of functions between two metric spaces in terms of open sets. The course will then cover two essential `topological' properties of metric spaces: connectedness and compactness. We will demonstrate that these properties are preserved under continuous mappings, thereby generalizing two fundamental theorems from elementary calculus—the Intermediate Value Theorem and the Extreme Value Theorem.
Satisfactory results in MATH4006-7.
Ability to read set notations.
Interests in learning and writing rigorous mathematical proofs.
Metric spaces (Definition of a metric, Examples)
Convergence of sequences (Definition, Examples)
Open and closed sets (Definition of open & closed balls, open & closed sets, limit points, interior & closure)
Continuous functions (three definitions : classical, sequential continuity and in terms of preimage of open sets)
Complete metric spaces (Cauchy sequences, Axiom of completeness of R, proof that C[a,b] is complete)
Contraction Mapping Theorem (Statement and proof)
Compactness (Definition, sequential compactness, compactness preserved by continuity, EVT)
Weierstrass Approximation Theorem (Statement and examples)
Connectedness (Connectedness and Path-connectedness, connectedness preserved by continuity, IVT)
Arzelà-Ascoli Theorem (Equicontinuity, statement and proof)