In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel \_{p}$.


Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$.



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> Assume that $S$ is a subvector space of $L^{q}[0, 1]$ such that the identity operator $\text{Id.}: (S, \parallel . \parallel \_{p}) \to (S, \parallel . \parallel \_{q})$ is a bounded operator. Does this implies that $S$ is a finite dimensional space?
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If I am note mistaken, this is proved for $p=2, q=\infty$, by Grothendieck.



A generalization of a theorem of Grothendieck
No. The condition implies that the subspace is isomorphic to a Hilbert space. In fact, Kadec and Pelczynski proved that a subspace of $L\_p$, $2<p<\infty$, is closed in $L\_r$ for some $r<p$ if and only if the subspace is isomorphic to a Hilbert space.


Kadec, M. I.; Pełczyński, A. Bases, lacunary sequences and complemented subspaces in the spaces Lp. Studia Math. 21 1961/1962 161–176.


Or look at the book by Albiac and Kalton.


