Is the only difference between functions just a change of basis?

[Monocles]

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).

 

[Tac-Tics]

Infinite dimensional spaces can get kind of weird, but some are easy enough to understand.

The "standard" basis for polynomials would be the set {1, x, x^2, x^3, ...}. Then, you can represent each polynomial with an infinite vector with a finite number of nonzero entries. For example, x^2+1 would be (1, 0, 1, 0, 0, ...) and 3x^3 - x^2 would be (0, 0, -1, 3, 0, ...). Add them together, and you get (1, 0, 0, 3, 0, ...) which is x^3 + 1.

Similarly with Fourier functions, you have the basis {1, cos x, sin x, cos 2x, sin 2x, cos 3x, sin 3x, ...}. You can represent these guys in the same way.

You can also do a change of basis. With polynomials, for example, we can use an ugly basis like {1, x + x^2, x - x^2, x^3 - x^4, x^3 + x^4, ...} or something like that.

But keep in mind that functionspaces don't play nicely all the time like R^n does. You can define the norm or "length" of a function by taking the integral over its domain. But you can only do this if you can guarantee that this integral is always finite! So for polynomials, you must restrict the domain to some bounded subset of R.