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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?

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Of course, an arbitrary collection of seminorms can be zero on some point $x_0$, without $x_0$ being zero.

However, Example.1.7. is about a very specific collection of seminorms:
$\mathcal{P} = \{p_{x^*}: x^* \in X^*\}$, 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.

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