We usually talk about $F[[x]]$, the set of formal power series with coefficients in $F$, as a topological ring. But we can also view it as a topological vector space over $F$ where $F$ is endowed with the discrete topology. And viewed in this way, $\{x^n:n\in\mathbb{N}\}$ is a Schauder basis for $F[[x]]$.
Now in contrast, $\{(x)_n:n\in\mathbb{N}\}$, where $(x)_n$ denotes the falling factorial, is not a Schauder basis for $F[[x]]$. That’s because if $\Sigma_na_n(x)_n$ never converges in the standard topology on $F[[x]]$ if infinitely many of the $a_n$’s are nonzero. But my question is, does there exist some alternate topology on $F[[x]]$ which makes $\{(x)_n:n\in\mathbb{N}\}$ a Schauder basis for $F[[x]]$ as a topological vector space over $F$ endowed with the discrete topology?
I don't see the point of the question.
With the axiom of choice there is some $F$-vector subspace such that $$F[[x]]=W\oplus F[x]$$
For a sequence $c_n\in F$ we get the unique decomposition $$\sum_{n\ge 0}c_n x^n= w(c)+\sum_{k=0}^{d(c)} a_k(c) x^k$$
When $c_n$ is zero for $n> N$ we have $w(c)=0$ thus $d(c)=N$,$a_k(c)=c_k$.
Sending $c_n$ to $$w(c)+\sum_{k=0}^{d(c)} a_k(c) (x)_k$$ gives an alternative topology on $F[[x]]$, the topology inherited from that of $F^\Bbb{N}$.
By definition of the alternative topology $$\lim_{N\to \infty} \sum_{n=0}^N c_n (x)_n =w(c)+\sum_{k=0}^{d(c)} a_k(c) (x)_k$$