Let $M$ be a closed subspace of LCS(locally convex space) $X$ with base field $\mathbb F$. Define $$\bar p_x (x^* +M^\perp ) := \inf_{m^* \in M^\perp}|(x^*+m^*)(x)|$$ for $x^* \in X^*$.
Then it can be shown $\{\ \bar p_x\}_{x\in X}$ is a separating family of seminorms, so that $(X^*/M^\perp , \{\ \bar p_x\}_{x\in X})$ is a LCS.
I want to show that the natural map $$\pi: (X^*,wk^*) \to (X^*/M^\perp , \{\ \bar p_x\}_{x\in X})$$ is an open map.
My attempt:
It suffices to show for any fixed $x_1,\dots x_n \in X$ and $\epsilon>0$, $$\pi\big(\bigcap_{i=1} ^n \{x^*\in X^*:|x^* (x_i)| < \epsilon\}\big)$$is open in the LCS $(X^*/M^\perp , \{\ \bar p_x\}_{x\in X})$.
I tried to show this by using the following Hahn-Banach theorem;
If $M$ is a closed subspace in a LCS $X$ and $x\notin M $, then for any $r \in \mathbb F\ $ there exists $m^* \in M^\perp$ s.t. $m^*(x)= r$.
I tried using the theorem to each $x_i \notin M$and adding the corresponding $m^*$'s, but could not proceed on.
Any help is appreciated.