Let $X$ be a subset of the Lebesgue space $L_2([0,1])$ satisfying the following:
For every $x \in X$ we have $0\le x\le \mathbf{1}$.
$X$ is norm compact as a subset of $L_\infty([0,1])$.
This means that $X$ is a norm compact subset of all $L_p([0,1)]$ for $1\le p\le \infty$.
We know that $L_2([0,1])$ is isomorphic to the sequence space $\ell_2$, and that every orthonormal basis of $L_2([0,1)]$ defines such an isomorphism. I'm interested in the image of $X$ under such isomorphisms. In particular the following:
Fix $0<p<2$. Under what additional conditions on $X$ does there exist an isomorphism $\Gamma_p\colon L_2([0,1]) \to \ell_2$ such that $$ \Gamma_p(X) \subseteq \ell_p\,. $$ The interest in particular is in the case where $p=1,1/2$.
Of course, the answer is affirmative if $X$ is finite dimensional. So the interest is in infinite dimensional sets.
