ON VARIOUS MODES OF SCALAR CONVERGENCE IN L_{0}(𝔛)

S. J. DILWORTH MARIA GIRARDI

Abstract. A sequence {f_{n}} of strongly-measurable functions taking values in a Banach space 𝔛 is scalarly null aė˙(resp. scalarly null in measure) if x^{∗}f_{n} → 0 aė˙(resp. x^{∗}f_{n} → 0 in measure) for every x^{∗} ∈ 𝔛^{∗}. Let 1 ≤ p ≤ ∞. The main questions addressed in this paper are whether an L_{p}(𝔛)-bounded sequence that is scalarly null aėẇill converge weakly aė˙(or have a subsequence which converges weakly aė˙), and whether an L_{p}(𝔛)-bounded sequence that is scalarly null in measure will have a subsequence that is scalarly null aė. The answers to these and other similar questions often depend upon p and upon the geometry of 𝔛.