Formal power-series substitution is the unique continuous k-algebra map

By [L1], every series $F={\sum }_{\alpha }{a}_{\alpha }{X}^{\alpha }$ has a convergent substituted sum $\varphi (F):={\sum }_{\alpha }{a}_{\alpha }{x}^{\alpha }\in A\mathrm{.}$ Finite truncations show that $\varphi$ respects addition and multiplication, and by construction $\varphi$ is a $k$-algebra map with $\varphi (X_i)=x_i$.

The map is continuous for the $(X_1,\dots ,X_n)$-adic topology on the source and the $\mathfrak{m}$-adic topology on the target, because every series all of whose monomials have total degree at least $N$ maps into ${\mathfrak{m}}^N$.

If $\psi$ is another continuous $k$-algebra map with $\psi (X_i)=x_i$, then $\psi$ agrees with $\varphi$ on the polynomial subring $k[X_1,\dots ,X_n]$. Every formal series is the limit of its polynomial truncations, and both maps are continuous, so they agree on all of $k\mathrm{⟦}{X}_1,\dots ,X_n\mathrm{⟧}$. Therefore $\varphi$ is unique.

Thus formal substitution is the unique continuous $k$-algebra map sending each indeterminate to the chosen maximal-ideal element.