Type II-ʌ-Weak Radon-Nikodym property in a Banach space associated with a compact metrizable Abelian group

Abstract

Let G be a compact metrizable abelian group with normalized Haar measure λ, Ґ the dual group of G and ʌ a subset of Ґ. Let X be a Banach space and f : G → X be a Pettis integrable function with respect to Ґ. It has been shown that the set {𝑓(𝛶):𝛶𝜀 ʌ} of the Fourier coefficients of 𝑓 is a relatively norm compact subset of X. We have shown by a counter-example that the converse of this result is not true, in general. We have introduced the idea of type II-ʌ-Weak Radon-Nikodym property (type II-ʌ-WRNP) of X and have shown that the converse is true for X having this property when ʌ is a Riesz set. We have also obtained several necessary and sufficient conditions for X to possess this property when ʌ is a Riesz set.