| I think that dirac delta solutions let us model point particles. We can lift differential operators onto the space of D' of distributions. This lets us extend the pde solution set with distributions. Conceptually this is how we would model a field with interacting particles using PDEs, and is as far as I know the reason we solve with distributions and consider them physical in the first place. The question of whether dirac is homotopic to vacuum would then mean exhibiting a homotopy in D', which is the dual space to the underlying function space, and represents both functions and distributions as functionals on a space of test functions. The topology of D' is the weak-* topology. Given a net N_t of distributions, N_t converges to N if N_t(g) converges to N(g) for all test functions g. So to make your proof tight, I propose we should lift it into D', where our family u_t of shrinking deltas is already a net. We'd then prove that u_t -> 0 converges for each test function. I think the result would be much more satisfying and watertight, since your current proof does not really make up its mind about what it's conclusion is. Now, I am not sure that the distributions need to be a hilbert spaces to solve PDEs in this sense. This is because distributions have derivatives and can be acted on by our lifted differential operators. These solutions are weak in the sense that they are solutions with respect to the test functions. But I also believe that the solutions are faithful, in the sense that any solution corresponding to a function in the base space will also be a solution for the original operator. Also, if you need our space of distributions to be a Hilbert space, we can pass to distributions over sobolev n space, the space of functions with square integrable nth derivatives. The distributions here end up being hilbert spaces! Not only that but the square integrability condition is pretty mild and still leaves us a lot of very useful functions for physics. I think that some quantum theories use these kinds of Hilbert spaces. Lol, take that with a grain of salt - this is lore to me, not terra firma. Anyhow, if you want to try to persuade yourself that we can't just throw away delta functions and other distributions, and then try to lift your proof into space of distributions, I think that would be very fun! I'll try to do that as well! Ps. Forgive me if I'm Tom Sawyering you or nerd sniping! Please also forgive me for any mistakes I'm making! |