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
Canonical name	BanachSpacesWithComplementedSubspaces
Date of creation	2013-03-22 16:02:59
Last modified on	2013-03-22 16:02:59
Owner	aube (13953)
Last modified by	aube (13953)
Numerical id	13
Author	aube (13953)
Entry type	Theorem
Classification	msc 46C15
Synonym	Lindenstrauss-Tzafriri theorem
Synonym	Lindenstrauss-Tzafriri complemented subspace theorem