In Axler's book he defines lists and says that its' lenght has to be finite
by definition each list has a finite length that is a nonnegative integer. Thus an object that looks like
(x1; x2; ...),
which might be said to have infinite length, is not a list
Later on, talks about $$F^{\infty} $$
So those aren't lists right, because the lenght is infinte, so what's that and what does F^inf means? All Functions that exists from N -> F? Whose element x_1 is the first one, x_2 is the second element and x_n is the nth element?
Your guess is correct. An infinite sequence like this is not a finite list, but can be regarded as a function $\mathbf N\to\mathbf F$. In the case of $(x_1,x_2,\ldots)$, if the function if $f$, then $f(1)=x_1$, $f(2)=x_2$, etc.