Banach-valued sequence spaces and tensor products

Let U be a reflexive Banach space with an unconditional basis and X be any Banach space. First, we gave sequential representations of the projective tensor product U X and the injective tensor product U X. Then through their sequential representations, we discuss some geometric properties of U X and U X, such as, reflexivity, the Rodon-Nikodym property, the analytic Radon-Nikodym property, being an Asplund space, being a Grothendieck space, weakly sequential completeness, and containing no copies of c₀, ℓ₁, and ℓ_∞. For example, we have obtained the following results: (1) U X has the (analytic) Radon-Nikodym property if and only if X does. (2) U X is an Asplund space if and only if X is an Asplund space and each continuous linear operator from U to X* is compact. (3) U X is a Grothendieck space if and only if X is a Grothendieck space and each continuous linear operator from U to X* is compact. (4) U X contains no copy of c₀ if and only if X contains no copy of c ₀; while, U X contains no copy of ℓ₁ if and only if X contains no copy of ℓ₁ and each continuous linear operator from U to X* is compact. (5) U X* contains a complemented copy of ℓ₁.; Finally, replacing U by some concrete Banach spaces, such as, classical Banach spaces ℓ_p and L_p[0,1], Orlicz spaces ℓ_M and L_M[0,1], and Lorentz spaces d( w, p) and L_w,p[0,1], we obtain characterizations of these geometric properties of the projective and the injective tensor products of these concrete Banach spaces with any Banach space.