On the Radon-Nikodym property and the Lewis-Radon-Nikodym property for a tensor norm

The Radon-Nikodym property and the Lewis-Radon-Nikodym property for tensor norms are introduced and discussed. In particular, it is shown that the Hilbertian tensor norm It introduced in [6, Section 3] has the Lewis-Radon-Nikodym property but does not have the Radon-Nikodym property. Instead, if alpha is a tenser norm that has the Lewis-Radon-Nikodym property, then it holds that alpha/ has the Radon-Nikodym property. However, we single out another of Grothendieck's natural tensor norms, namely the projective tenser norm A, that does have the Radon-Nikodym property. Furthermore, it is shown that if alpha is a tenser norm with the Radon-Nikodyni property, then \alpha and /alpha have the property as well, but in general alpha\ need riot have the property. However, both tensor norms gamma(p) and gamma(p)\ are shown to have the Lewis-Radon-Nikodym property. Furthermore, it is deduced that the tenser norms gamma(p)/, h/ and \h/ have the Radon-Nikodym property. We also bring on board the least of things: we show that the injective tensor norm V does riot have the Lewis-Radon-Nikodym property.