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Good $\ell_2$-subspaces of $L_p$, $p>2$. (English) Zbl 1194.46014
In a recent preprint, {\it R. Haydon, E. Odell} and {\it Th. Schlumprecht} [“Small subspaces of $L_p$,” \url{arXiv:0711.3919}] show that a Hilbertian subspace of $L_p$, $p>2$, contains a further subspace $Z$ that is $(1+\varepsilon)$-isomorphic to $\ell_2$ and complemented in $L_p$ by a projection of norm $\le (1+\varepsilon)\gamma_p$, where $\gamma_p$ is the $L_p$-norm of a standard Gaussian random variable. Their proof uses random measures and types à la Krivine and Maurey. Here, the author gives another proof that avoids these means and depends only on a version of the central limit theorem for martingales.

46B09Probabilistic methods in Banach space theory
46B25Classical Banach spaces in the general theory of normed spaces
46E30Spaces of measurable functions
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