Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice

TL;DR

This paper explores the impact of the Axiom of Choice on functional analysis, highlighting potential issues and differences that arise when the axiom is not assumed, aimed at analysis students.

Contribution

It provides an accessible overview of how the Axiom of Choice influences key results in functional analysis for students.

Findings

Illustrates effects of omitting the Axiom of Choice in functional analysis

Highlights potential failures and issues without the axiom

Educational focus on conceptual understanding

Abstract

This text is meant for analysis students who want to learn more about the effects of the axiom of choice on functional analysis, and the things that may go wrong in its absence. As this is a text aimed for analysis students, we will not focus on the set theoretic proofs or dwell on any particular set theoretic assumptions that are needed in order to prove certain results.