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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
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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 1364 times total.

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