An Elementary Expository Study: From Metric Spaces to Hilbert Spaces
Ismail Gemaledin, Iusuf Gemaledin
TL;DR
This paper provides an elementary, accessible exposition of metric spaces and the topologies they induce, tracing how distance, convergence, and continuity give rise to open and closed sets and to broader topological structure. It develops the core notions of metrics and pseudo-metrics, introduces metric and topological equivalence, and demonstrates how different metrics can yield the same topology through concrete examples on finite and infinite-dimensional spaces. It then connects subspaces, bounded metrics, and product constructions to build up to the infinite-dimensional Hilbert space $H=\{u=(u_i):\sum_i u_i^2<\infty\}$, embedding $\mathbb{R}^n$ as subspaces and proving fundamental inequalities like Cauchy–Schwarz. The results establish foundational tools for analysis and topology and provide a coherent path from elementary metric spaces to the structure of Hilbert spaces with explicit, constructive examples.
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.