A question of member of Mathoverflow:
In this question the norm of is denoted by .
Let and be two arbitrary real numbers with .
> Assume that is a subvector space of such that the identity operator is a bounded operator. Does this implies that is a finite dimensional space?
If I am note mistaken, this is proved for , by Grothendieck.
The answer of Bill Johnson on MO:
In fact, Kadec and Pelczynski proved that a subspace of , $2<p<\infty$, is closed in for some 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.