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Good ${\ell }_{2}$-subspaces of ${L}_{p}$, $p>2$. (English) Zbl 1194.46014
In a recent preprint, R. Haydon, E. Odell and 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 $\left(1+\epsilon \right)$-isomorphic to ${\ell }_{2}$ and complemented in ${L}_{p}$ by a projection of norm $\le \left(1+\epsilon \right){\gamma }_{p}$, 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.
##### MSC:
 46B09 Probabilistic methods in Banach space theory 46B25 Classical Banach spaces in the general theory of normed spaces 46E30 Spaces of measurable functions