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A well known result (is it by Pelczynski?) says that the $\ell_p$ spaces for $1\leq p<\infty$ are $\ell_p$-saturated. This means that every infinite dimensional subspace of $\ell_p$ contains a further subspace isomorphic to $\ell_p$. But what happens in $L_p(\mathbb{R}^n)$ or $L_p[0,1]$ spaces?

Question: Is it true that every infinite dimensional subspace of $L_p$ contains a further subspace isomorphic to $L_p$?

What I know for sure is a nice dichotomy theorem by Kadets-Pelczynski which states that every infinite dimensional subspace of $L_p[0,1]$ is either isomorphic to $\ell_2$ or contains a further subspace which is isomorphic to $\ell_p$. Additionally, by Khinchine's theorem, every $L_p[0,1]$ space contains a subspace isomorphic to $\ell_2$.

If the answer to the question was positive, then this would imply that every $L_p$ space could be embedded in $\ell_2$. This may be a contradiction, but I can't say for sure.

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No, they are not self-saturated as by Kchinchine's inequality, each $L_p$ contains a copy of $\ell_2$ and every subspace of $\ell_2$ is isomorphic to $\ell_2$. $L_p$ is not isomorphic to $\ell_2$ because it contains $\ell_p$. That $\ell_p$ and $\ell_2$ ($p\neq 2$) are not isomorphic follows from Pitt's theorem.

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