I just started teaching myself a little bit about Hilbert spaces and functional analysis in general, and I had an idea that I don't think I've seen in my textbook, so I'm wondering if I have the right idea: All functions "live in" the space of all functions. When you are doing something like decomposing a function as a power series, or a fourier series, is the only thing you are doing a change of basis? So, is the only difference between, say, e^x and 1 + x + (x^2)/2! + (x^3)/3! + ... is the basis you have written the function in? If so, in what basis are typical elementary equations usually written in? Does it have a name? How do you even define the basis the functions are in without "knowing" what the function is (since the basis itself is made up of functions)? Any other additional tidbits on this topic are appreciated as well. Try to keep things simple though, I've only taken 1 proof based math class (though it was on vector spaces, but we never got to Hilbert spaces).