## unconditional bases for $(\sum^\infty_{n=1} \oplus \ell^n_2 )_{\ell_1}$ [closed]

are there unconditional bases for $(\sum^\infty_{n=1} \oplus \ell^n_2 )_{\ell_1}$ ?

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Actually, if I understand it correctly, the notation is fine. It just seems to me that the answer should be "yes, in the obvious way". There is an obvious candidate for an unconditional basis, have you tried to show that it works? – Yemon Choi Mar 27 2012 at 7:00
If the space is $\lbrace (x_{n,m})_{m\le n}: \sum_n \left( \sum_m |x_{n,m}|^2\right) ^{1/2} <\infty\} \rbrace$ then the "unit matrices" $e_{n,m}$ (all entries $0$ except at the $(n,m)$-th spot) form an unconditional basis. – Jochen Wengenroth Mar 27 2012 at 7:16
is there more than one [up to equivalence] normalized unconditional basis ? – Rafael Mar 27 2012 at 17:48
The $\ell_1$ sum of infinite dimensional Hilbert spaces has a unique (up to permutations and equivalence, of course) semi normalized unconditional basis; see Bourgain, J.(B-VUB); Casazza, P. G.(1-MO); Lindenstrauss, J.(IL-HEBR); Tzafriri, L.(IL-HEBR) Banach spaces with a unique unconditional basis, up to permutation. Mem. Amer. Math. Soc. 54 (1985), no. 322, iv+111 pp. 46B15 – Bill Johnson Apr 1 2012 at 12:14