Assembled topology and quotient

Let ${\mathcal{T}}_n$ be the locally convex inductive limit over all compact supports and closed cones inside $G_n$. Larger cones remove Fourier seminorms, so the stage inclusions are continuous. Give ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ the locally convex direct-sum topology ${⨁}_{n\ge 0}{\mathcal{T}}_n$. Give formal series the product topology of their coefficients and ${\mathcal{E}}_H$ the quotient topology.