Talk:Banach-Alaoglu Theorem

I will try to write down a proof this evening, if I find the time. - Giallo (30.1.13, 12:28)


 * Awesome; that'd be greatly appreciated! --Lord_Farin (talk) 11:31, 30 January 2013 (UTC)

I have written a first (much incomplete) outline of the proof. I'll have to work a bit on the proof for the two lemmas, since my source (Royden & Fitzpatrick) is a bit vague about them, but they shouldn't be too hard. Unfortunately I have a couple of exams soon, thus I'll leave it to you until then. By the way, i don't know if this is really the theorem of Banach-Alaoglu or if it should only be named after Alaoglu. This one was proven by Alaoglu, while Banach has shown that if $X$ is separable, then the closed unit ball in $X^*$ is weak* sequentially compact. - Giallo (30.1.13, 21:39)


 * Conway lists it as Alaoglu's Theorem. --Lord_Farin (talk) 21:28, 30 January 2013 (UTC)


 * Ok, I have moved it to the page for Alaoglu's theorem (the discussion, too). - Giallo (30.1.13, 22:45)

I also have written down the true (at least, I hope) Banach-Alaoglu theorem with proof. - Giallo (30.1.13, 23:26)

Retread
As presented in (whether this is accurate or not):
 * in a dual Banach space, the unit ball is weak star compact, and more generally, the polar of a neighborhood of the origin in a topological vector space is also weak star compact.

Whoever takes this on is invited to explore and document all of the redlinks in the above, and to establish whether the above is as fully general as possible, and then if possible to bring it back down to the instance of the real number line (or complex if this is appropriate) and so provide an instance for those for whom the intricacies of Banach spaces are too far over their heads. Like (until I care to do the work to catch up with all this) over mine. --prime mover (talk) 12:54, 20 October 2022 (UTC)


 * Can anyone explain what the polar of a neighborhood of the origin means? --Usagiop (talk) 14:27, 20 October 2022 (UTC)


 * The polar of $U$ is the set $\set {f \in X^\ast : \cmod {\map f x} \le 1 \text { for each } x \in U}$. Caliburn (talk) 20:25, 21 February 2023 (UTC)

I intended to put up a third red-link free proof of this tomorrow, but there is a problem. The general statement (which I will prove for NVSs, and probably come back for TVSs. There obviously are no balls in general TVSs so the statement and proof is very different) states that the unit ball in $X^\ast$ is $w^\ast$-compact. The assumption that $X$ is separable implies that the unit ball in $X^\ast$ with the $w^\ast$ topology is metrizable (this is another theorem that I will be proving - it's also an iff), so sequential compactness is equivalent to compactness. So you get sequential compactness in the separable case, but not in general. How should I deal with this? Caliburn (talk) 20:25, 21 February 2023 (UTC)


 * Sorry but what is the problem? --Usagiop (talk) 21:41, 21 February 2023 (UTC)


 * These proofs assumed that $X$ is separable and make a stronger conclusion. The proof I'm going to put in has weaker conditions and makes a "weaker" conclusion. (well, compactness and sequential compactness don't generally imply each-other but ygm) I could relegate this current page to a corollary, I guess. Separable case and general case? I don't know how heavily these proofs use separability, I haven't really looked at them in detail yet. Caliburn (talk) 22:05, 21 February 2023 (UTC)