Is there a Banach space with an unconditional basis which contains an isomorphic copy of $\ell_p$ (for some $p\in (1,\infty)$) but such that no copy of $\ell_p$ is complemented?
Copies of $\ell_p$ in spaces with unconditional basis
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functional-analysis
banach-spaces
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0Do you have such an example for spaces without an unconditional basis? – 2012-10-04
1 Answers
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The answer is yes; a good reference for what follows is Chapter 6 of Albiac and Kalton's book Topics in Banach space theory (see in particular p.130 and p.160).
The Banach spaces $L_p[0,1]$, $1
P.S. With regards to the question posed in Theo's comment above, the answer is also yes in that case by considering $L_1[0,1]$, which does not have an uncondtional basis (p.144 of Albiac-Kalton) and does not have any infinite-dimensional reflexive complemented subspaces (since it has the Dunford-Pettis property).