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Let be a projective system of Banach vector bundles whose limit is a Frechet bundle of fibre type F. We construct a generalized bundle of frames P(£·) o f E by revising entirely the classical notion and by substituting GL(F) with an appropriate enlarged structure group. This is imposed by the pathology of GL(F), which renders meaningless the ordinary frame bundle. As a result, we prove that E is associated with P (£ ) and linear connections of E correspond to (principal) connections of P(£). In particular, the former are necessarily projective limits of connections on the bundles Et.

A Generalized Frame Bundle for Certain Fréchet Vector Bundles and Linear Connections