Are Banach Manifolds intrinsically interesting? 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. 
 A: I am of two minds on this topic. It is much easier to work on  Banach manifolds because the implicit function theorem on such spaces    has a simple formulation.  On the other hand,   as the examples of gauge theory or    the theory of pseudo-holomorphic curves show, in these contexts one works   not with one Banach manifold, but with several, determined by    stronger and stronger Sobolev  norms. One  important  part of the game  is to   conclude that objects with a priori weaker Sobolev regularity are   in fact smooth.  This feels   very much like we are implicitly working on a    Frechet manifold.
One draw back of Banach spaces is that they do not have  many smooth functions on them, and the notion of  real analycity on such spaces is problematic.          Let's take the example of Seiberg-Witten equations.    These are quadratic equations  in its variables, so intuitively they ought to be real analytic, though I do not know how to formulate this  rigorously in a Sobolev context.
Why do I care  about real analycity? In the real analytic  context one can formulate an intersection theory involving not necessarily smooth objects. For example, the  point $0\in\mathbb{R}$ is a solution  of the quadratic  equation $x^2=0$. It is a degenerate zero, and from the point of view  of intersection theory it   has multiplicity $0$.  My hope  is  that this real analytic point of view would allow one to deal with mildly degenerate solutions of the the Seiberg-Witten equations,  and assign multiplicities   to such solutions. 
A: In his remarkable thesis Douady proved that, given a compact complex analytic space $X$,  the set $H(X)$ of analytic subspaces of $X$ has itself a natural structure of analytic space .
If $X=\mathbb P^n(\mathbb C)$ for example, then $H(X)$ is the Hilbert scheme  $ Hilb(\mathbb  P^n(\mathbb C))$.
However the problem is much more difficult for non algebraic $X$.
Douady solved it by massive use of Banach analytic manifolds, the most important of them being the grassmannian of complemented closed subspaces of a Banach space. 
The thesis starts with the candid statement of its aim: "Le but de ce travail est de munir son auteur du grade de docteur-ès-sciences mathématiques et l'ensemble H(X) des sous espaces analytiques compacts de X d'une structure d'espace analytique", that is to endow its author with the title of doctor in mathematics and the the set H(X) of compact analytic subspaces of X with the structure of analytic space.
