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 $l\_{p}$. 



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> **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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Subspaces of $L_{p}(2<p<\infty)$
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 $p\le r \le 2$.


