Let $p>2$. Let $X$ be a subspace of $L_{p}$. Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$. My question is: if $X$ is isomorphic to $l_{2}$, does $X$ contain a subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$?

Thank you!

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 $l_{2}$?

Let $p>2$. Let $X$ be a subspace of $L_{p}$. Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$. Haydon, Odell and Schlumprecht proved that if $X$ is isomorphic to $l_{2}$, then, for every $\epsilon>0$, $X$ contains a complemented subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$. For the second case, if $X$ contains a subspace isomorphic to $l_{p}$, does $X$ contains a subspace that is $(1+C_{p})$-isomorphic to $l_{p}$? where the constant $C_{p}$ depends only on $p$.

Thank you!

Let $p>2$. Let $X$ be a subspace of $L_{p}$. Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$. My question is: if $X$ is isomorphic to $l_{2}$, does $X$ contain a subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$?

Thank you!