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Banach spaces with complemented subspaces

(Theorem)

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).

"Banach spaces with complemented subspaces" is owned by aube.

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Other names:

Lindenstrauss-Tzafriri theorem, Lindenstrauss-Tzafriri complemented subspace theorem

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Cross-references: topology, induces, structure, Hilbert space, isomorphic, complemented, subspace, closed, Banach space, theorem
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This is version 10 of Banach spaces with complemented subspaces, born on 2006-06-27, modified 2008-04-12.
Object id is 8100, canonical name is CharacterizationOfAHilbertSpace.
Accessed 2609 times total.

Classification:

46C15 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Characterizations of Hilbert spaces)

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