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 “not containing inverses” 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 Krein–Mil’man–Rutman theorem on stability of minimal systems [4] […]

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

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 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.)