| Let $V$ be a Banach space, such that for each closed subspace $M$ there exists a closed subspace $N$ such that $M\cap N=0$ and $M+N=V$ (i.e. every closed subspace is complemented). Then $V$ is isomorphic to a Hilbert space (i.e. there exists a Hilbert space structure on $V$ that induces the original topology on $V$ as a Banach space). |
Let $V$ be a Banach space, such that for each closed subspace $M$ there exists a closed subspace $N$ such that $M\cap N=0$ and $M+N=V$ (i.e. every closed subspace is complemented). Then $V$ is isomorphic to a Hilbert space (i.e. there exists a Hilbert space structure on $V$ that induces the original topology on $V$ as a Banach space). |
