Let T be a locally compact Hausdorff space, called base space. Suppose for each t in T there is a (real or complex) Banach space Et. A vector field x is an element in the product space t T Et, that is, x (t) Et, for all t T.

We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an into isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.

For Banach lattices X and Y, let X | | Y and X | | Y denote the positive projective and injective tensor products of X and Y, respectively. In this paper, we characterize the inheritance of reflexivity and containment of copies of c 0, 1, by X | | Y and X | | Y from X and Y, when one of them is an atomic Banach lattice. In this case, we also give an affirmative answer to an open question of Jeurnink.