Representation of by vector-valued function spaces

In this paper we use results from the theory of tensor products of Banach spaces to establish the isometry of the space of so-called strongly p-integrable functions in a Banach space X to the space of integral X-valued operators on Lp(μ) and the complete projective tensor product of Lp(μ) with an arbitrary Banach space X. Although there are similar results in recent papers (for instance in [Q. Bu, P.K. Lin, Radon–Nikodym property for the projective tensor product of Köthe function spaces, J. Math. Anal. Appl. 293 (2004) 149–159] and [J. Diestel, J.H. Fourie, J. Swart, The projective tensor product II: The Radon–Nikodym property, Rev. R. Acad. Cienc. Ser. A Mat. 100 (2006) 75–100]), our contribution in the present paper is to remove all restrictions attached to the Banach space X. Through similar techniques, the result is then also considered in the context of the complete projective tensor product of a Banach lattice X (satisfying some conditions) with an arbitrary Banach space Y.