Good \(\ell_2\)-subspaces of \(L_p\), \(p>2\) (Q967143)

In a recent preprint, \textit{R.\,Haydon, E.\,Odell} and \textit{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 \(\leq (1+\varepsilon)\gamma_p\), 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.