Some geometric proprties of a sequence space related to ℓp

Abstract

The sequence space m(φ), introduced and studied by W.L.C. Sargent in 1960, is closely related to the space ℓp. In this paper we obtain an explicit formula for the Hausdorff measure of noncompactness of any bounded subset in m(φ). We also show that m(φ) enjoys the weak Banach-Saks property, while C(m(φ)) = 2. This shows that the condition C(X) < 2, known to be sufficient for the space X to have the weak Banach-Saks property, is not a necessary one.