# Banach spaces with complemented subspaces

Theorem. [Lindenstrauss-Tzafriri]

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).

Title Banach spaces with complemented subspaces BanachSpacesWithComplementedSubspaces 2013-03-22 16:02:59 2013-03-22 16:02:59 aube (13953) aube (13953) 13 aube (13953) Theorem msc 46C15 Lindenstrauss-Tzafriri theorem Lindenstrauss-Tzafriri complemented subspace theorem