Seminorms ported by a compact

Question

Let $X$ be a Banach space, and $U\subset X$ an open balanced set. A seminorm $\rho$ in $\mathcal{H}(U)$ is said to be ported by a compact $K\subset U$ if for all open set $V$ such as $K\subset V \subset U,$ there is a constant $c(V)$ with $$\rho(f)\leq c(V)\lVert f\rVert_V$$ for all $f \in \mathcal{H}(U).$ The topology generated by this family of seminorms is known as Nachbin compact-ported topology $\tau_\omega$. What we need to prove is: if $\rho$ is a $\tau_\omega$-continuous seminorm in $\mathcal{H}(U)$, then $$\rho^*(f)=\sum_{n=0}^\infty \rho(P_n)$$ also defines a $\tau_\omega$,continuous seminorm, where $f=\sum P_n$ represents the power series around $0$. I already proved that $\rho^*$ is always finite: if $K \subset U$ is a compact set such as $\rho$ is ported by $K$, there is a $\varepsilon>0,$ and $\lambda>1$ with $$\lambda(K+B_\varepsilon(0))\subset U$$ and also $\lVert f\rVert_{\lambda(K+B_\varepsilon(0))}<\infty$. Note that, by Cauchy inequality, for all $x\in K+B_\varepsilon(0)$ $$\lvert P_n(x)\rvert\leq \lambda^{-n}\sup_{\lvert \xi\rvert=\lambda}\lvert f(\xi x)\rvert\leq \lambda^{-n} \lVert f\rVert_{\lambda(K+B_\varepsilon(0))},$$ so, choosing $V=K+B_\varepsilon(0)$ we have $$\rho^*(f)=\sum_{n=0}^\infty \rho(P_n)\leq c(V)\sum_{n=0}^\infty \lVert P_n\rVert_{V}\leq c(V)\lVert f\rVert_{\lambda V} \sum_{n=0}^\infty \lambda^{-n}<\infty.$$ I guess we need to do something similar for proving $\rho^*$ is $\tau_\omega$-continuous.