# Subspaces of $L_{p}(2<p<\infty)$

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 $l_{2}$?

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 $p\le r \le 2$.
• 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 $l_{2}$? – Dongyang Chen Jun 30 '16 at 0:26
• 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. – Gideon Schechtman Jun 30 '16 at 8:30