Topologically finitely generated Hilbert -modules

For a Hilbert C ( X ) -module V , where X is a compact metrizable space, we show that the following conditions are equivalent: (i) V is topologically finitely generated, (ii) there exists K ∈ N such that every algebraically finitely generated submodule of V can be generated with k ≤ K generators, (iii) V is canonically isomorphic to the Hilbert C ( X ) -module Γ ( E ) of all continuous sections of an (F) Hilbert bundle E = ( p , E , X ) over X , whose fibres E x have uniformly finite dimensions, and each restriction bundle of E over a set where dim E x is constant is of finite type, (iv) there exists N ∈ N such that for every Banach C ( X ) -module W , each tensor in the C ( X ) -projective tensor product V ⊗ π C ( X ) W is of (finite) rank at most N .