In the introduction to 'A convenient setting for Global Analysis', Michor & Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open subsets of the modeling space for many modeling spaces."

But finite-dimensional manifolds are found to be interesting even though they can be embedded in some Euclidean space (of larger dimension). (Actually this seems to me, to make the above claim intuitively plausible, so that claim should be no more than we should expect).

But they do go on to say that "Banach manifolds are not suitable for many questions of Global Analysis, as ... a Banach Lie group acting effectively on a ﬁnite dimensional smooth manifold it must be ﬁnite dimensional itself.", which does seem a rather strong limitation.

studyinginfinite dimensions and actually proving anything because there you have the implicit function theorem. In particular if you have an elliptic problem, e.g. holomorphic curves, it's much easier to think of the solution space inside a Banach manifold of maps and then use elliptic regularity it prove it's really inside the smooth locus. – Jonny Evans Mar 9 '12 at 7:09