Hi there, Ive been asking in the Quantum Physics Forum about some foundational stuff and Ive got stuck in why we assume in QM that the states space is linear. I have not found any deeper answer than "because it fits with observation". So Im now trying the other way around. I believe that the state space should at least be a metric space, so, is there any theorem or something that says something like: "If we add a Metric Space properties A, B an C then the space is Linear" I mean, Im looking something like those theorems that state which are the things we have to add to a topological space in order to make it metrizable (that is, to find the only metric that is coherent with its topology). In this line of thought, Id like to find some theorems that state which are the properties that we have to add to a Metric Space in order to make it "Linearizable" (that is, to find the only field, like R or C or Q or Quaternions or Grassmann numbers or whatever, that is coherent with its metric). Does this exist? Thanks!