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Let $p>2$ and $X$ a subspace of $L_{p}$.

Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$.

Question: if $X$ is isomorphic to $l_{2}$, does $X$ contain a subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$?

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Yes. That follows, e.g., from the Krivine-Maurey theory of stable spaces even if it was known before their work. For $p<2$ you get from their theory, and more or less classical considerations, Aldous' theorem that every subspace of $L_p$ contains for every $\epsilon > 0$ a subspace that is $1+\epsilon$-isomorphic to $\ell_r$ for some $p\le r \le 2$.

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  • $\begingroup$ Let $1<p<2$. Let $X$ be a subspace of $L_{p}$ that contains no isomorphic copy of $l_{p}$. Does $X$ contain for every $\epsilon>0$ a subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$? $\endgroup$ – Dongyang Chen Jun 30 '16 at 0:26
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    $\begingroup$ No. $L_p$ contains a subspace isometric to $\ell_r$, for every $p<r<2$. This was first observed by M. Kadec and follows easily from the existence and properties of $r$ stable random variables, due to P. Levy. $\endgroup$ – Gideon Schechtman Jun 30 '16 at 8:30

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