An example of a family of seminorms but not norms.

Question

The topology of LCTVS can be induced by a family of seminorms $\{p_\lambda\}_{\lambda \in \Lambda}$.

My question: can $\{p_\lambda\}_{\lambda \in \Lambda}$ satisfy the following condition?

for any $\lambda$, there exists non-zero $x$(depending on $\lambda$), such that, $p_\lambda(x) = 0$?

An example of such a family of seminorms would be great!

Answer 1

On $C(\mathbb R)$ consider the semi-norms $p_n(f)=\sup \{|f(x)|: |x| \leq n\}$. Then for each $n$ there is a non-zero continuous function $f$ which is $0$ on $[-n,n]$ so $p_n(f)=0$.