Back in the fall semester of 2015-2016 I had taken a course in functional analysis. One of the reasons I wanted to take that course (other than needing the credits to finish my Ph.D.) is that I was always curious about the functional analytic results related to the axiom of choice, and my functional analysis wasn’t strong enough to sift through these papers.
I was very happy when the professor, Matania Ben-Artzi, allowed me to write a final paper about the usage of the axiom of choice in the course, instead of taking an exam.
I have decided to finally post this paper online. It covers some possible disasters in functional analysis without the axiom of choice, or with “seemingly nice” assumptions (such as automatic continuity). You can find it in the Papers section.
My goal was to make something readable for analysts, rather than to provide a retread of older set theoretic proofs (and some model theoretic proofs). So some things are left with only a reference, and other set theoretic statements are formulated in a rather unusual way. If you are interested, I’d be happy to hear any remarks on this paper, or suggestions for improvements in the comments below or over email. If you know analysts that might be interested to read this, please let them know of the existence of the paper.
For the set theorists the paper can be seen as a nice historical overview of these results, and perhaps it can be of use in other ways.