Talk:Contractibility of unit sphere in Hilbert space

cool
This article is so cool. �Preceding unsigned comment added by 195.176.0.55 (talk) 18:49, 9 May 2008 (UTC)

new article Cgwaldman 17:11, 16 October 2007 (UTC)

I'm not too happy with the most recent set of changes to this article.

All of the explanatory comments in the proofs were removed, making the article shorter and more streamlined, but harder to understand (for example, the fact that T maps S to its equator was removed). Is the goal to make articles terse, or to make them informative? Cgwaldman 20:35, 31 October 2007 (UTC)


 * hi, Cgwaldman. thanks for contributing this page. hope we see you around. let me respond to your comments. i don't think the current version of the article is terse. elaborating too much on trivial claims actually make the article more opaque.


 * consider, for instance, the second section. the key features of the proof are that the shift is an isometry and has no eigenvalues, therefore one has a curve from z to Tz that lies entirely on S. the argument in fact works with any powers of T, therefore any nonsurjective isometry. the pretty obvious fact that the unit sphere in Hilbert space can be mapped (linear) isometrically onto its own equator is not essential here. similarly, "there is no point z on the sphere S for which T(z) is the antipode � z of z" follows from that T has no point spectrum. ditto for "If z lies on S, then this path does not pass through the origin".


 * as for the last section, which particular basis one chooses for L2[0,1] is again irrelevant. the statement: "...two functions are considered to be identical if they agree on all but a countable set of points." is not quite right. to mention that measurable functions need not be continuous is distracting from the argument. also, it's misleading to say that the proof is less elementary and uses more functional analysis (in fact really no functional analysis is used) than the previous argument. Mct mht 02:14, 1 November 2007 (UTC)

The topological argument is wrong
The statement ``Since Sk is compact, so is its image under ³. Hence the image is contained in some n-dimensional subspace of H.'' in the first proof is FALSE!!!! I think the unit sphere in Hilbert space may not be homeomorphic to a CW-complex, in which case this proof cannot be salvaged. Whitehead's theorem does NOT apply.71.111.220.102 (talk) 01:26, 26 December 2008 (UTC)

Thanks for the commend, but I'm pretty sure the Hilbert sphere is a CW-complex. CW-complexes are defined inductively, and you can form the Hilbert sphere by attaching 2 k-cells to S(k-1) for each k>2, and taking the union of all of these spaces. Maybe I'm missing something but it seems that provides a good CW-complex structure. When I get a chance, I'll dig up a copy of this paper: S. Kakutani, Topological properties of the unit sphere in Hilbert space, Proc. Imp. Acad. Tokyo 19 (1943), 269-271. which should settle this question. Cgwaldman (talk) 06:14, 2 January 2009 (UTC)


 * Nope, infinite-dimensional Hilbert spaces are not CW-complexes (at least, not as an inductive limit over finite-dimensional subspaces), pretty much exactly for the reason you give. Consider the Hilbert space ℓ2 from the Hilbert space article.  Now consider the sequence $$z_n = 1/n$$, this is in ℓ2, and consider the subset of ℓ2 that consists of all sequences which agree with $$z_n$$ for the first k terms, and the remaining terms are zero.  This is a bounded countable set, so its closure is compact (with limit point the sequence $$z_n$$).  But the closure isn't contained in a finite-dimensional subspace of ℓ2 since it contains a countable-infinite linearly independant set.  Normalize these vectors if you want everything to be in the unit sphere.  Rybu (talk) 06:37, 20 November 2009 (UTC)


 * The comment beginning with "Nope, infinite-dimensional Hilbert spaces are not CW-complexes" may be correct in its conclusion, but this comment is not a proof of that fact. (The comment shows only that a certain countable subset of ℓ2 does not lie in a finite-dimensional subspace.)2600:1700:E1C0:F340:F498:2C87:73A6:6C70 (talk) 19:38, 8 July 2018 (UTC)