This paper is concerned with
projective and inductive tensor products of bundles of Banach spaces. Let π : E → S
and ρ : F → T be bundles of Banach spaces over the locally compact Hausdorff
spaces S and T, with fibers {Es: s ∈ S} and {Ft: t ∈ T}, respectively. Let Γ0(π)
and Γ0(ρ) be their spaces of sections which disappear at infinity. We show the
existence of a bundle π⊗ρ : E⊗F → S ×T whose fibers are {Es⊗Et: (s,t) ∈ S ×T};
if σ ∈ Γ0(π) and τ ∈ Γ0(ρ), then their pointwise tensor σ ⊙ τ defined by
(σ ⊙ τ)(s,t) = σ(s) ⊗ τ(t) is a section of the bundle π⊗ρ : E⊗F → S × T. Further,
we show the existence of a bundle π⊗ρ : E⊗F → S × T whose fibers are
{Es⊗Ft: (s,t) ∈ S × T}, and demonstrate that Γ0(π)⊗Γ0(ρ) and Γ0(π⊗ρ) are
isometrically isomorphic.
