I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should especially contain $\mathbb{R}$ and $\mathbb{C}$.


If e.g. $f \circ g = h$ holds for formal power series $f, g, h \in R[[X]]$, I want to be able to conclude $f(g(r)) = h(r)$ for all $r \in R$ for which both sides of the equation converges in $R$.


For those structures $R$ it is obviously necessary that the Cauchy product of two convergent series (if it converges) equals the ordinary product. I know that this always is the case for series over the complex numbers, but does this also remain true in a more abstract setting?



Lifting identities of formal power series
Take $g(x)=x-x^2\in\mathbb{Z}[[x]]$. There is an $f(x)\in x\mathbb{Z}[[x]]$ which is an inverse to $g(x)$ under composition (this is true because $g(x)$ has constant term 0 and linear term a unit; we could also write down $f(x)$ explicitly). We have $g(1)=0$, and $f(0)=0$ (both expressions converge in any topological ring, as they have only finitely many non-zero terms), so the expression $f(g(1))$ converges, and equals 0. However, $h(x):=f(g(x))=x$, and $h(1)=1\neq 0$. This construction works in every topological ring, so the statement "$h=f\circ g\Rightarrow h(r)=f(g(r))$ for all $r\in R$ for which both sides converge" does not hold in ANY non-zero topological ring $R$.


