  By replacing the category of smooth vector bundles over a manifold with the
category of what we call smooth Euclidean fields, which is a proper enlargement
of the former, and by considering smooth actions of Lie groupoids on smooth
Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie
groupoids. The notion of smooth Euclidean field we introduce here is the
smooth, finite dimensional analogue of the usual notion of continuous Hilbert
field.
