Banach spaces with complemented subspaces
Theorem. [Lindenstrauss-Tzafriri]
Let be a Banach space, such that for each closed subspace there exists a closed subspace such that and (i.e. every closed subspace is complemented). Then is isomorphic to a Hilbert space
(i.e. there exists a Hilbert space structure on that induces the original topology
on as a Banach space).