**Edit:**  This question has been significantly revised.

Some recent developments in computational geometry (for example see http://geometry.stanford.edu//papers/fmfrmbs-obsbg-12/fmfrmbs-obsbg-12.pdf) are based on the idea of considering the pullback of a map between two manifolds.  As the pullback is a linear map (insofar that it is well-defined), it is much more friendly to work with than the original map, and many tools from Hilbert spaces can now be exploited.  However, the "price paid" is that now you are perhaps working in a higher dimensional space, or some information about the original map is lost.

I am very interested to learn of any other mathematics literature that may be related to these ideas or this approach.  Thank you very much.