Functionals on locally convex space of complex polynomials

Question

Let $\mathcal{Z}=\{z_n\}_{n=1}^{\infty}$ be an infinite subset of complex numbers and $\mathfrak{P}:=\{\varphi_z \ : \ \varphi_z(p):=|p(z)|\}_{z\in\mathcal{Z}}$ a separating family of seminorms defined on $\mathbb{C}[x]$. How to prove that any linear bounded functional on $(\mathbb{C}[x],\tau)$ is of the form $\sum\limits_{j=1}^{n}\alpha_j p(z_j)$ for some $\alpha_1,...,\alpha_n\in\mathbb{C}, \ z_1,...,z_n\in\mathcal{Z}$, where $\tau$ is a locally convex topology defined by $\mathfrak{P}$ ?

Answer 1

Continuity of a linear functional $f$ on $\mathbb C[x]$ means that $|f|\le C \max\lbrace \varphi_z(f): z\in E\rbrace$ for some finite set $E$. For the linear functionals $f_z$ defined by $f_z(p)= p(z)$ you thus have $\bigcap\limits_{z\in E}$ kern$(f_z) \subseteq$kern$(f)$ and this implies that $f$ is a linear combination of the $f_z$.