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Zbl 1194.46014
Alspach, Dale E.
Good $\ell_2$-subspaces of $L_p$, $p>2$. (English)
[J] Banach J. Math. Anal. 3, No. 2, 49-54, electronic only (2009). ISSN 1735-8787/e

In a recent preprint, {\it R.\,Haydon, E.\,Odell}\/ and {\it Th.\,Schlumprecht} [``Small subspaces of $L_p$,'' 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.
[Dirk Werner (Berlin)]

MSC 2000:: *46B09 Probabilistic methods in Banach space theory
46B25 Classical Banach spaces in the general theory of normed spaces
46E30 Spaces of measurable functions

Keywords: subspaces of $L_p$; well complemented subspace; Hilbertian subspace; central limit theorem

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