an:05702385 Zbl 1194.46014 Alspach, Dale E. Good $\ell_2$-subspaces of $L_p$, $p>2$. EN Banach J. Math. Anal. 3, No. 2, 49-54, electronic only (2009). ISSN 1735-8787/e 2009

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*46B09 46B25 46E30 subspaces of $L_p$; well complemented subspace; Hilbertian subspace; central limit theorem 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 \`a 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)