Metric Spaces

It is recommenced to have done an a course on Calculus and Real analysis before taking this course. Other than that, basic algebra is assumed.

This contains basic inequalities that needs to be known for this course and are used in various proofs.

• Inequalities

This chapter notes contains notes on the basics of metric spaces, some special metric spaces used throughout the course and some important results regarding metric Also introduces the notion of sub-spaces and isometries

• Basics of Metric Spaces

Introduces notion of open balls, closed ball, open sets, closed sets and various definitions and properties.

• Topology of Metric Spaces

This chapter generalizes the notion of sequences and convergence from \(\mathbb{R}\) to an abstract metric space. We then define further properties of sequences like the Cauchy Property, Completeness, etc.

• Sequences in Metric Spaces

This chapter generalizes the notion of continuity from \(\mathbb{R}\) to an abstract metric space and give 5 equivalent definitions of continuity. Finally we look at contractions and the famous Contraction Mapping Theorem

• Continuity

Introduces notions of Compactness and Sequential compactness. Also generalizing Bolzano Weirstrass Theorem and Heine Borel Theorem.

Introduces the notion of connectedness