Transfinite dimensions

TL;DR

This paper explores the properties of countable subsets of orthonormal bases in Hilbert spaces, establishing their equivalence with linear bases and providing a simplified proof of the Erdős-Kaplansky theorem.

Contribution

It demonstrates the equipollence of countable basis subsets with all linear bases in Hilbert spaces and offers a more natural proof of the Erdős-Kaplansky theorem.

Findings

Countable subsets of orthonormal bases are equipollent with all linear bases.

Provided a simplified, more natural proof of the Erdős-Kaplansky theorem.

Established foundational properties of transfinite dimensions in Hilbert spaces.

Abstract

Let H be a Hilbert space and let F be the family of all countable subsets of an orthonormal basis of H. We show that if F is infinite then F is equipollent with every linear basis of the vector space H. In doing so we also present a short proof of the Erd\"os-Kaplansky theorem more natural and much easier than the one by Bourbaki.