Proof that a $\mathbb C[[h]]$-module homomorphism $V[[h]]\to W[[h]]$ is automatically $h$-adically continuous.

Question

Consider the $\mathbb C[[h]]$-module $V[[h]]$, consisting of formal power series in the indeterminate parameter $h$ with values in $V$, where $V$ is some complex vector space. Define $W[[h]]$ similarly.

I read in several places that when one has a module homomorphism $\phi:V[[h]]\to W[[h]]$, it is automatically continuous with respect to the $h$-adic topologies. How does one prove the statement?

Answer 1

Let $\phi:V[[h]]\to W[[h]]$ be a $\mathbb C[[h]]$-module homomorphism, i.e., $\phi$ is $\mathbb C[[h]]$-'linear'. Open neighborhood bases at $x\in V[[h]]$ and $\phi(x)\in W[[h]]$ for the $h$-adic topologies on $V[[h]]$ and $W[[h]]$, respectively, are given by $\mathcal U = \{x+h^nV[[h]]\,:\,n\in\mathbb N\}$ and $\mathcal V = \{\phi(x)+h^nW[[h]]\,:\,n\in\mathbb N\}$, by definition of the $h$-adic topology. Now for any open neighborhood $\phi(x)+h^nW[[h]]\in\mathcal V$ we can pick the open neighborhood $x+h^nV[[h]]\in\mathcal U$, and then $\mathbb C[[h]]$-linearity, together with the obvious fact that $\phi(V[[h]])\subset W[[h]]$, implies that $\phi(x+h^nV[[h]]) = \phi(x)+h^n\phi(V[[h]])\subset \phi(x) + h^n W[[h]]$. This shows that $\phi$ is continuous at $x$, which is an arbitrary point.