Let X be an infinite dimensional Banach space. The paper proves the non-coincidence of the vector-valued Hardy space Hp([ ], X) with neither the projective nor the injective tensor product of Hp([ ]) and X, for 1 < p < ∞. The same result is proved for some other subspaces of Lp. A characterization is given of when every approximable operator from X into a Banach space of measurable functions [Fscr ](S) is representable by a function F:S → X as x [map ] 〈F(·), x〉. As a consequence the existence is proved of compact operators from X into Hp([ ]) (1 [les ] p < ∞) which are not representable. An analytic Pettis integrable function F:[ ] → X is constructed whose Poisson integral does not converge pointwise.