An Elementary Expository Study: From Metric Spaces to Hilbert Spaces

TL;DR

This paper provides an accessible introduction to metric spaces, their topologies, and the relationships between different metrics, aimed at undergraduate students to deepen understanding of fundamental concepts in analysis and topology.

Contribution

It offers an elementary exposition connecting metric spaces to their topologies, emphasizing examples, equivalence of metrics, and their role in analysis and topology education.

Findings

Clarifies the basic axioms of metric spaces

Explains how different metrics can induce the same topology

Provides illustrative examples for undergraduate learning

Abstract

Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an elementary exposition of metric spaces and their associated topologies. We start by recalling the basic axioms through which we understand a metric and examine various examples. The induced topology is next discussed with emphasis on open and closed sets, continuity and limits. In addition, we compare equivalent metric spaces and illustrate how different metrics can generate but the same topological structure. The presentation is designed to be easy to follow and accessible to undergraduate students, by combining classical definitions with illustrative examples that allow a deeper understanding of the aforementioned concepts.