# If a collection of seminorms on some point $x_0$ is zero then $x_0$ is zero?

The following is Example.1.7. page 101 Functional Analysis book of Conway:

1.7. Example. Let f be a normed space. For each $$x^*$$ in $$X^*$$, define $$p_{x^*}(x) = |x^*(x)|$$. Then $$p_{x^*}$$ is a seminorm and if $$\mathcal{P} = {\{p_{x^*}: x^* \in X^*}\}$$, $$\mathcal{P}$$ makes $$X$$ into a LCS.

1.2. Definition. A locally convex space (LCS) is a topological vector space (TVS) whose topology is defined by a family of seminorms $$\mathcal{P}$$ such that $$\cap_{p \in \mathcal{P}} {\{x: p(x) = 0}\} = (0)$$.

I don't understand Example.1.7. : Does it state that if a collection of seminorms on some point $$x_0$$ is zero then $$x_0$$ is zero? If so how this claim is true?

Of course, an arbitrary collection of seminorms can be zero on some point $$x_0$$, without $$x_0$$ being zero.
$$\mathcal{P} = \{p_{x^*}: x^* \in X^*\}$$, where, for each $$x^*$$ in $$X^*$$, we define $$p_{x^*}(x) = |x^*(x)|$$.
It is easy to prove that, for each $$x^*$$ in $$X^*$$, $$p_{x^*}$$ is in fact a seminorm.
Moreover, by Hahn-Banach, for each $$x \in X$$, there is $$x^*$$ in $$X^*$$, such that $$x^*(x)=\|x\|$$. So we have that, there is $$p_{x^*}\in \mathcal{P}$$, such that $$p_{x^*}(x) = |x^*(x) |= \|x\|$$ So, if, for all $$p \in \mathcal{P}$$, $$p(x) = 0$$, then $$x=0$$. So, $$\cap_{p \in \mathcal{P}} {\{x: p(x) = 0}\} = (0)$$.
So, $$\mathcal{P}$$ makes $$X$$ into a LCS.