On the isomorphism between Banach spaces and Hilbert spaces

Lindenstranss et al. (1971) proved that a Banach space X is isomorphic to a Hilbert space if every closed subspace in X is complemented. In this paper a different characterization is given, i.e. a Banach space X is isomorphic to a Hilbert space iff there is a linear bounded positive operator mapping X into X^*. According to this new characterization, the positiveness condition, which has often been used in the optimal control problem in Hilbert spaces to guarantee the existence, cannot be simply assumed in Banach spaces