# The Axiom of Choice and Self-Dual Vector Spaces

I have uploaded a note titled The Axiom of Choice and Self-Duality of Vector Spaces. Here is a short summary and background.

It is a well known fact (in $\ZFC$ at least) that if $V$ is a vector space, and $V^\ast$ is the algebraic dual of $V$ then $V\cong V^{\ast\ast}$ if and only if $\dim V<\infty$.

Some long time ago, after a discussion with Pete L. Clark on math.SE, I set about finding a counterexample. After giving up at first, I found about automatic continuity. It turns out that in models like Solovay’s model every linear functional from a Banach space to the field (real numbers or complex numbers) is automatically continuous. In such model, if so, any reflexive Banach space is also isomorphic to its double dual.

I began writing a short note with all the relevant theorems and information, hoping to find my first publication there, but alas as I was finishing I saw that this result (and more) was already covered in Schechter’s immense book Handbook of analysis and its foundations. Now that I have a website, it seems like a good reason to make final adjustments and upload the note.

This is my first note, and any comment or suggestion will be most helpful for future notes (which are coming, I can assure you).

Summary:

In models where every set of real numbers has the Baire property it turns out that every linear operator from a Banach space to a separable normed space is automatically continuous. In particular every linear functional is automatically continuous, and therefore the algebraic dual and the topological dual are the same.

Let $V$ be a vector space over $\mathbb R$. Denote by $V^\ast$ the algebraic dual of $V$, and for a topological vector space denote by $V^\prime$ the continuous dual. We will show that the following implications are not provable without the axiom of choice:

1. $V\cong V^\ast$ implies that $\dim V<\infty$;
2. $V\cong V^{\ast\ast}$ by a natural isomorphism if and only if $\dim V<\infty$;
3. If $V$ is a Banach space, $V^\prime$ is reflexive if and only if $V$ is reflexive;
4. If $V$ is a reflexive Banach space, $W\subseteq V$ is a closed subspace, then $W$ is also reflexive;
5. If $V^\prime$ is separable then $V$ is separable.

The Axiom of Choice and Self-Duality of Vector Spaces

### 5 Responses to The Axiom of Choice and Self-Dual Vector Spaces

1. That’s nice. Have you considered convertin this to HTML? or markdown (using Pandoc)? Then you could post it in its full glory :)

• Well, I am not a huge fan of reading things off a website. I am more inclined towards using a PDF reader (even on my iPhone!). Seeing how it’s only two and a half pages, I might consider doing that. But generally I don’t find this as an appealing option for the moment.

• Well, a real alternative would be an epub3 file, but that’s after you’ve got some html…

• I’m not sure I see the problem with .pdf files. It is, after all, a portable document [format]. Had you said that ten years ago when I was fighting my computer for every bit of free RAM I might had understood that, but nowadays? I don’t… I suppose this is a whole other discussion. :)

• Ha, yes, I remember those day :) Yes, this is a whole other discussion and a long overdue post of mine…