The Strong Property (B) for $$L_p$$ Spaces
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AbstractGiven a purely non-atomic, finite measure space $$(\Omega ,\Sigma ,\nu )$$ ( Ω , Σ , ν ) , it is proved that for every closed, infinite-dimensional subspace V of $$L_p(\nu )$$ L p ( ν ) ($$1\le p<\infty $$ 1 ≤ p < ∞ ) there exists a decomposition $$L_p(\nu )=X_1\oplus X_2$$ L p ( ν ) = X 1 ⊕ X 2 , such that both subspaces $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic to $$L_p(\nu )$$ L p ( ν ) and both $$V\cap X_1$$ V ∩ X 1 and $$V\cap X_2$$ V ∩ X 2 are infinite-dimensional. Some consequences concerning dense, non-closed range operators on $$L_1$$ L 1 are derived.
1973 ◽
Vol 25
(2)
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pp. 252-260
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1960 ◽
Vol 97
(2)
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pp. 254-254
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1992 ◽
Vol 53
(1)
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pp. 9-16
1979 ◽
Vol 31
(2)
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pp. 441-447
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1977 ◽
Vol 24
(2)
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pp. 129-138
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1985 ◽
Vol 8
(3)
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pp. 433-439