Sometimes I think I have a handle on Banach spaces. Then I contemplate the example of Tsirelson space, which is a Banach space that does not contain as a subspace any classical sequence space (*c _{0}* or

*l*).

_{p}Sometimes I think I have a handle on Banach spaces. Then I contemplate the example of Tsirelson space, which is a Banach space that does not contain as a subspace any classical sequence space (*c _{0}* or

I don’t understand the definition on that wikepedia page. Clearly, the space is constructed as a subset of some sequence space. But which one? Lâˆž perhaps? Also, property 2 in the definition references a set m which is nowhere defined.

(I don’t know if this site handles HTML and Unicode in comments, and there is no preview facility. So I’ll take my chances. My apologies if an unreadable comment results.)

Just following up on my previous comment: Unicode OK, HTML tags stripped.

Harald: These spaces are usually constructed by starting with c_{00}, defining a norm on this space, and then completing. c_{00} is just the vector space of all sequences which are eventually zero. One could work inside l_\infty if you wanted, but that is a little confusing, as the sup norm is not important.

I agree with property 2 in the wikipedia article makes no sense. Perhaps m should be A? Indeed, this is a very weird way to try to define the norm. Usually the norm of T is defined inductively: one defines a norm \|.\|_1, then uses this to define another norm \|.\|_2, and so forth. The norm on T is then the limit of \|.\|_n.

Checking out the “talk” section shows that Tsirelson himself has dropped by, and isn’t too happy with the page!

I didn’t understand the construction in the Wikipedia article either, but I thought that was just a result of my own ignorance. If the article is actually bad, then let’s fix it! We can thrash out the details either here or in the discussion page there.

I found Tsirelson’s 1974 paper. It’s doi:10.1007/BF01078599 (on Springerlink) for those at an institution with a valid subscription (a green document icon should show if you have full access).

Tsirelson employs the same mystical m without explanation, but it is clear that it must be a sequence space. Further, from Lemma 2 in the paper it becomes obvious that m should be c_0. What the Wikiepedia article calls block-disjoint is called â€œnot containing inversesâ€ in Tsirelson’s paper, except that he expressly allows the final vector to have infinitely many nonzero elements. In the resulting space, the sequence {e_j} is an absolute basis, and the dual sequence is an absolute basis for the dual. After a superficial reading, the paper looks accessible enough. It’s only three (big) pages plus four references. My biggest reservation when it comes to actually understanding the proof is this reference:

[…] we apply the Kreinâ€“Mil’manâ€“Rutman theorem on stability of minimal systems [4] […]

[4] M. G. Krein, D. P. Mil’man, and M. A. Rutman, â€œOn a property of a basis in a Banach space,â€, Zapiski Khar’k Matem. Ob-va, 16, 106â€“110 (1940).

Heck, I went and edited the wikipedia page. It could still need more work, but I don’t think I am particularly qualified. But this one I felt I could do. And while I was at it, I created a page for uniformly convex spaces.

Hmm: it appears that my understanding of the Tsirelson space is actually its dual. I learnt about these things from the following book: Google Print page called “Classical Sequences in Banach Spaces” (or “Spates” as Google Print calls it, in case you need to the search again). Do a search for “Tsirelson” in the book, and you’ll find on pages 63 and 64 (which are both accessible from the search, but not from browsing: you know how Google Print works) the inductive construction of a norm I was talking about. This seems easier to understand to me, but technically it doesn’t give you Tsirelsonâ€™s original space, but the dual (which has the same properties of course). I might edit the Wiki page if I get bored with my own research this afternoon…