Strongly embedded subspaces of p-convex Banach function spaces.

Let $$X(\mu )$$ be a p-convex ( $$1\le p<\infty $$) order continuous Banach function space over a positive finite measure $$\mu $$. We characterize the subspaces of $$X(\mu )$$ which can be found simultaneously in $$X(\mu )$$ and a suitable $$L^1(\eta )$$ space, where $$\eta $$ is a positive finite measure related to the representation of $$X(\mu )$$ as an $$L^p(m)$$ space of a vector measure $$m$$. We provide in this way new tools to analyze the strict singularity of the inclusion of $$X(\mu )$$ in such an $$L^1$$ space. No rearrangement invariant type restrictions on $$X(\mu )$$ are required. [ABSTRACT FROM AUTHOR]