Some Properties of the Injective Tensor Product of Banach Spaces

Abstract

Let X and Y be Banach spaces such that X has an unconditional basis. Then X $$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ \otimes } $$ Y , the injective tensor product of X and Y , has the Radon–Nikodym property (respectively, the analytic Radon–Nikodym property, the near Radon–Nikodym property, non-containment of a copy of c 0, weakly sequential completeness) if and only if both X and Y have the same property and each continuous linear operator from the predual of X to Y is compact.