The Levi-Civita connection in infinite dimensions Is there an analogue of the Fundamental Theorem of Riemannian Geometry for (some subclass of) infinite-dimensional manifolds?

The Koszul formula works for the uniqueness part of this theorem, but not for the existence part in the infinite dimensional case due to the lack of isomorphism between co-tangent and tangent spaces. 
With a proper adjustment of definitions one can find analogous facts. See p. 489 A. Kriegl, P.W. Michor, The Convenient Setting of Global Analysis, e.g. here, and also in Andrew Stacey's notes here and here.
