Let $X$ be a topological vector space. Let $X^\ast$ denote its continuous dual. It is possible to endow $X^\ast$ with the weak star topology:
Def.1: If $e_x: X^\ast \to \mathbb C$ is the map $\varphi \mapsto \varphi (x)$ then endow $X^\ast$ with the initial topology with respect to $\{e_x\}_{x\in X}$.
Def.2: Denote by $p_{x}: X^\ast \to \mathbb R$ the map $\varphi \mapsto |\varphi(x)|$. Endow $X^\ast$ with the topology induces by these seminorms. As I understand it means the topology with basic open set of the form $$U_{\varphi,x,\varepsilon} = \{ \psi \in X^\ast: |\varphi(x)-\psi (x)|<\varepsilon\}$$
Now if instead $X^\ast$ is endowed with the initial topology with respect to $\{p_x\}_{x\in X}$, what sort of topology does one obtain? Does it have a name?