Course Overview: The theory of metric spaces is foundational to much of pure and applied mathematics, for it axiomatises the notion of a sensible distance defined on a set and builds a rich theory surrounding this. The development of this theory forms the central part of the course. Before and after this though, we consider two related topics of a slightly different nature. We begin the course by extending the idea of differentiability from one to several real variables. This is somewhat more subtle than one might expect, and of central importance in real analysis. And we end the course by studying some aspects of the geometry of the complex plane. Some of the topics considered will be touched on again in the companion course Complex Analysis, which runs in parallel to this one.
Learning Outcomes: Students will have become familiar with the central concepts and results in metric spaces, and thus some of the basic ideas of point-set topology. They will have a deeper understanding of real differentiation, by developing it in higher dimensions, and will have greater appreciation of the beautiful geometry of the complex plane.
Course Synopsis: Real differentiation in \(\R^2\). Continuous partial derivatives imply differentiability and differentiability implies continuity. Some topology on \(\R^2\). Continuous functions on compact sets achieve their maximum. Different examples of distance. The statement (no proof) of the inverse function theorem on \(\R^2\). [4]
The definition of a metric space. Examples of metric spaces in various parts of mathematics. Norms, and metrics derived from a norm on a real vector space, particularly \(\ell^1, \ell^2, \ell^{\infty}\)-norms on \(\R^n\). Metrics on product spaces. Open and closed balls, and bounded sets. Limits, continuity, and uniform continuity (\(\varepsilon\)-\(\delta\) definition). Function spaces, of continuous real-valued functions and of bounded functions on a metric space. Isometries and homeomorphisms. Open and closed sets and their basic properties. Continuity in terms of pre-images of open and closed sets. A subset of a metric space inherits a metric. Interiors, closures and limit points. [3]
Completeness (but not completion). Completeness of the space of bounded real-valued functions on a set, equipped with the supremum norm, and the completeness of the space of bounded continuous real-valued functions on a metric space. Lipschitz maps and contractions. Contraction Mapping Theorem. [2.5]
Connected metric spaces, path-connectedness. Closure of a connected space is connected, union of connected sets is connected if there is a non-empty intersection, continuous image of a connected space is connected. Path-connectedness implies connectedness. Connected open subset of a normed vector space is path-connected. [2]
Definition of sequential compactness and proof of basic properties of sequentially compact sets. Preservation of sequential compactness under continuous maps, equivalence of continuity and uniform continuity for functions on a sequentially compact set. Equivalence of sequential compactness with being complete and totally bounded. Open cover definition of compactness. Statement of Heine-Borel for \(\R^n\) and proof that compactness implies sequential compactness (statement of the converse only). [2.5]
The Riemann sphere and the isometry with the extended plane via stereographic projection. Möbius transformations: definition and examples. Conformal maps. [2]
