Resumen de Type II-L-weak Radon-Nikodym property in a Banach space associated with a compact metrizable abelian group

N. D. Chakraborty, Sk. Jaker Ali

• Let G be a compact metrizable abelian group with normalized Haar measure ¸, ¡ the dual group of G and ¤ a subset of ¡. Let X be a Banach space and f : G ¡! X be a Pettis integrable function with respect to ¸. It has been shown that the set f ^ f(°) : ° 2 ¤g of the Fourier coe±cients 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-¤-Weak Radon-Nikodym property (type II-¤-WRNP) of X and have shown that the converse is true for X having this property when ¤ is a Riesz set. We have also obtained several necessary and su±cient conditions for X to possess this property when ¤ is a Riesz set.