I seem to be missing something. I’m reading this as looking for functions f and g from R to R, such that f(x)+g(x)=1 and f(x+p)=f(x).

If I were reading that correctly, then for any periodic f, g(x)=1-f(x) would do, meaning any old periodic function would do.

What an I missing?

Am I also reading it right that the author is saying the axiom of choice is equivalent to the statement that every vector space has a finite basis? I don’t get how that allows infinite dimensional vector spaces. If not, and it’s just that every vector space has a basis that’s maybe infinite, then what’s the justification of the Hamel basis being finite?

I feel like I’m missing a lot between the lines here.