SMALL BANACH BUNDLES AND MODULES

We characterize those continuously-normed Banach bundles E → X with compact Hausdorff base whose spaces Γ(E) of global continuous sections are topologically finitely-generated over the function algebra C(X), answering a question of I. Gogić and extending analogous work for metrizable X. Conditions equivalent to topological finite generation include: (a) the requirement that E be locally trivial and of finite type along locally closed and relatively Fσ strata in a finite stratification of X; (b) the decomposability of arbitrary elements in ℓ^p(Γ(E)), 1 ≤ p < ∞ as sums of ≤ N products in ℓ^p(C(X)) · Γ(E) for some fixed N; (c) the analogous decomposability requirement for maximal Banach-module tensor products F ⊗ _C(X₎Γ(E) or (d) equivalently, only for F = ℓ¹(C(X)).