## Weak Cauchy sequences in $$L^ 1(E)$$.(English)Zbl 0579.46025

Let E be a Banach space, and let $$(f_ n)$$ be an equi-integrable sequence of $$L^ 1(E)$$. Then there is a sequence $$g_ n\in co(f_ n,f_{n+1},...)$$ such that, for almost each $$\omega$$, either the sequence $$(g_ n(\omega))$$ is weak-Cauchy or it is equivalent to the unit vector basis of $$\ell^ 1$$. As application, we characterize weakly precompact sets of $$L^ 1(E)$$ and show that $$L^ 1(E)$$ is weakly sequentially complete whenever E has this property. Extensions are given to the space of measures of bounded variations.

### MSC:

 46E40 Spaces of vector- and operator-valued functions 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46A50 Compactness in topological linear spaces; angelic spaces, etc.
Full Text: