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

Talagrand, Michel

Am. J. Math. 106, 703-724 (1984).

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.

Keywords:

equi-integrable sequence; weakly precompact sets; weakly sequentially complete