The Strong Property (B) for $$L_p$$ Spaces

Abstract Given 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 $$L_p(\nu L p ( $$1\le p<\infty $$ 1 ≤ < ∞ ) there exists decomposition )=X_1\oplus X_2$$ = X ⊕ 2 such both subspaces $$X_1$$ and $$X_2$$ are isomorphic to $$V\cap X_1$$ V ∩ infinite-dimensional. Some consequences concerning dense, non-closed range operators on $$L_1$$ derived.