I don't know if it counts for "analysis", but toposes (=topoi) appear to be [related][1] to quantum mechanics.
There's even a [book][2] on the subject.
For a specific example, a keyword might be [Bohr topos][3].


  [1]: http://ncatlab.org/nlab/show/topos+approach+to+quantum+mechanics
  [2]: http://www.springer.com/physics/quantum+physics/book/978-3-642-35712-1
  [3]: http://ncatlab.org/nlab/show/Bohr+topos