In Serge Lang's Foundamentals of Differential Geometry, the definition of vector bundles was developed in Banach spaces in stead of $\mathbb{R}^n$. It's not too hard to find some examples on finite dimensional manifolds, but for infinite dimensional case, I found it difficult to get one on the Internet, wondering if I missed something important. Is there anyone can share some infinite dimensional examples? Indeed the tangent bundle of any Banach manifold has already been an example, but is there any other example which is more concrete? I mean, is it possible to give the total space and bundle projection different from tangent bundle?
This wiki page tells people exactly what I mean by vector bundle modeled on Banach spaces. Any suggestions/examples/hints appreciated! Thank you!