A classical result of Kadec and Pelczynski

A question of member of Mathoverflow:

In this question the norm of L^{P}[0,1] is denoted by \|.\|_p.

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

> Assume that S is  a subvector space of L^{p}[0, 1] \bigcap L^{q}[0, 1] such that the identity operator \text{Id.}: (S, \|.\|_p) \to (S, \|.\|_q) is a bounded operator.  Does this implies that S is  a finite dimensional space?

If I am note mistaken, this is proved for p=2, q=\infty, by Grothendieck.

The answer of Bill Johnson on MO:

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.

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