Grothendieck spaces and duals of injective tensor products

Let E and F be Fréchet spaces. We prove that if E is reflexive, then the strong bidual (E ⊗̂εF)″b is a topological subspace of Lb(E′b, F″b). We also prove that if, moreover, E is Montel and F has the Grothendieck property, then E ⊗̂εF has the Grothendieck property whenever either E or F″b has the approximation property. A similar result is obtained for the property of containing no complemented copy of c0.