Recently I had a very odd thought. Can we do calculus on functions that take and return functions (eg, )?
Well, we’d need to have a conceptual idea of space between functions. Well, we could borrow one from the real numbers by constructing a trivial set of functions that are isomorphic to the real numbers: .
Great. So we have an organized set of functions. It seems appropriate to call this a function space. Now we just need a functions that acts on it so we can do calculus.
Because it would be ironic, let’s choose as our function.
Hm… what happens when we apply to functions of the form ? .
Blast it! That’s outside our function space! What will we do? We’ll add another function space .
The cartesian product of those two earily similar to the second degree real polynomial space, … Let’s consider them to be the same thing!
So, it seems reasonable to say that the Jacobian of the -dimensional version of this is:
Just some random thoughts. I mentioned this to a math professor and he said some stuff had been done on it in the field of functional analysis, but a quick search didn’t show much…