Posted in Analysis and Optimization

# 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.

> 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 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.