> Yes, I implied infinite dimensions is useful. Linear algebra in QM tends to be infinite-dimensional. You don't have to appeal to QM for their usefulness. Fun

> Yes, I implied infinite dimensions is useful. Linear algebra in QM tends to be infinite-dimensional. You don't have to appeal to QM for their usefulness. Function spaces are important in applied and computational mathematics because of their use in understanding integral equations (the theory of integral equations before Hilbert spaces was a giant mess), partial differential equations, calculus of variations, approximation theory, etc. No doubt marshallp will respond that this is all bullshit because it ultimately boils to finite processes running on finite state machines, where there are no infinite sets in sight, let alone infinite dimensions. The equivalent approach to physics would be an extreme form of empiricism, banning the use of concepts like electrons (as some logical positivists actually proposed to do in the early twentieth century) and requiring all physical laws to be stated in terms of directly observable phenomena, whatever that means.