According to the book by Huybrechts, Complex Geometry: An Introduction, this is a nice space and may be regarded, after some form of completion, as an infinite-dimensional manifold. How is this done, i.e. how is this space constructed as a manifold? I'd like some references.
I know how to construct the space of almost complex structures on a real even-dimensional vector space and show that it is a complex manifold. I would prefer a construction that departs from this point.