Necessity of Axiom of Choice in Functional Analysis given ZF + Dependent Choice

What do we get with the Axiom of Choice (AC) in Functional Analysis that cannot be accomplished with Zermelo-Fraenkel (ZF) plus the Axiom of Dependent Choice (DC)?

So, for instance, just dusted off and opened my old class notes and saw

1. Hahn-Banach (see note below)
2. Baire Category (see note below)

3. Open Mapping / Closed Graph thrms

4. Uniform Boundedness Principle
5. Projection Lemma (Hilbert Space)
6. Unit Ball is weak-* compact (Banach-Algaoglu)
7. Riesz Representation Theorem (see note below)
8. Spectral Theorem
9. Separation of Convex sets by Hyperplanes
10. Basic theorems about distribution functions/L^P spaces

... and any other theorems you can think of that fit in this theme of being well known to anyone who took a basic course in functional analysis, and useful to people who use analysis in their work. Please add to list b/c I am sure I forgot some theorems.

(note on Hahn-Banach: In the spirit of the question, the most concrete form that is still abstract enough to use in the proofs of the other theorems is fine here. For instance, in my notes I have $p$ sublinear on a N.V.S. $X$ and $V$ a subspace of $X$, $f\in V^*$ and $|f|\le p$ on $V$ as the assumptions.

note on Baire Category: The version that every complete metric space (or Banach space) is a Baire space would likely be sufficient here since I believe that the second one that locally compact Hausdorff spaces are Baire might not be standard material for a basic course in Functional Analysis. Please correct me if I'm wrong!)

So, which require AC if one already accepts ZF+DC?

Edit This is intended to be more a question about the internal logical dependencies of Functional Analysis than about logic and set theory. A good answer does not need to prove each of the theorems separately. In particular, One might show theorems $n_1$ and $n_2$ can be proven with DC by citing good links. Then say "a standard proof for $n_3$ uses $n_1$ and some epsilon delta stuff. A standard proof $n_4$ uses $n_2$ and $n_3$ plus image of compact sets is compact, so also doesn't need full AC." For the ones that do need AC, maybe a good link for one and then a link showing that others are equivalent under ZF.

In other words, what I am looking for is for someone to take the few theorems about DC vs. AC that have already been proven, and flesh this out to the rest of (basic) functional analysis by discussing logical dependencies within the field of functional analysis.

Please only assume a background in functional analysis, not in Foundations (sets/logic/etc. beyond everyday use). References to other questions where the details have been worked out in more rigour are quite sufficient.

• Look like you've got at least eight good questions, of the form "is this theorem provable in ZF+DC or is some stronger form of choice needed". However, this site prefers one post per question. – bof Oct 28 '16 at 1:43
• @bof Asking eight separate questions is not the intent or goal of the post. As the first sentence states its to find out exactly where in functional analysis the axiom of choice is necessary. I recall doing the proofs and it showed up everywhere. But back then I did not know of other options like the Dependent Choice. Without clarifying the scope of the question, though, it is likely that the answers would ignore many parts of functional analysis that are important and familiar to the readers. The only other option I can see is erase the list, but I don't see how this improves the question – cantorhead Oct 28 '16 at 2:27
• @bof Also, please keep in mind that these examples are very interrelated. I very much expect to find that answering the question for one or two of the proposed examples will immediately answer many of the remaining examples. – cantorhead Oct 28 '16 at 2:35
• This question is somewhat broad. I have written a small review paper last year for a course in functional analysis. While it doesn't fully cover your list, you might find it useful nonetheless. karagila.org/2016/… – Asaf Karagila Oct 30 '16 at 12:30
• Not to mention that in general, many of the proofs of "necessity" will have to resort to very deep knowledge in set theory and mathematical logic. So saying "Hey, about you give me a bunch of proofs, without assuming any knowledge in set theory and foundations" is really a bit of an overly broad request here. You should also look into two books: Eric Schechter's "Handbook of Analysis and its Foundations" as well as Horst Herrlich's "The Axiom of Choice". Possibly also Fremlin's "Measure Theory vol. 5" which has choiceless results. – Asaf Karagila Oct 30 '16 at 12:33

2 Answers

There is an obvious item missing from your list: the existence of nonmeasurable sets. The second Solovay model (see here) is a model of ZF+DC and all of its sets are Lebesgue measurable. Without being able to mention the existence of nonmeasurable sets it becomes difficult to motivate the machinery of $\sigma$-algebras. Furthermore Terry Tao has pointed out some nifty applications of nonmeasurable sets, particularly in the context of Robinson's framework.

In another direction, you can't even start talking about the Stone-Čech compactification, e.g., $\beta\mathbb N$, unless you have something stronger than DC.

My guess would be that the existence of an invariant subspace of a polynomially compact operator on Hilbert space cannot be established without a stronger form of choice. Certainly Lomonosov's proof, while simple, uses powerful fixed-point results that most likely rely on AC.

• I left this out on purpose because I consider it a topic in Analysis rather than Functional Analysis. Some textbooks on real and functional analysis (see Serge Lang's book, for instance) do the entirety of Functional Analysis before considering measure. Others (see Royden) break it up the other way. The only caveat here is that I did mention L^p spaces. This might be a clue that ZF+DC is not sufficient for L^p space theory. – cantorhead Oct 30 '16 at 13:18
• Again, this might be an issue of how different people organize subject matter in their head, but for me Stone-Czech compactification is purely topological. The Baire Category theorem is also purely topological, but I included it because it is used so heavily in introductory Functional Analysis texts. – cantorhead Oct 30 '16 at 13:54
• @cantorhead OK well you can't claim the invariant subspace problem is not part of functional analysis without getting into trouble with Halmos :-) – Mikhail Katz Nov 1 '16 at 12:18
• There is a definite need to limit the scope of this question. I chose to limit it by referring to "introductory", or "basic" functional analysis, but I also had in mind that part of the basis for adding a theorem to the list would be how widely known the theorem is and how widely used/applied is the theorem. From my talks with physicists and engineers, most people who use functional analysis can at least recall the names of these theorems, and many have used them along the way. Not being as familiar with the invariant subspace problem as I am with the other theorems, I am not in the best – cantorhead Nov 3 '16 at 21:31
• position to decide if the invariant subspace problem belongs in this list of "basic" theorems that everyone should know. If others agree then I will add it. Or you could just edit it in. – cantorhead Nov 3 '16 at 21:32

Cédric Villani's undergraduate lecture notes covering the content of Brezis's functional analysis book as well as Lebesgue integral and Fourier integral without the Axiom of Choice using "modern language" is 70% complete (about 400 pages) according to this interview (Google translate).