I am reading Conways A Course in Functional Analysis.
A locally convex space is a topological vector space whose topology is defined by a family of seminorms $\mathcal{P}$ such that $\bigcap_{p \in P} \{x:p(x)=0 \}=(0)$.
I was looking for an example of a topological vector space that is not a locally convex space.
I found the following: Why isn't $\ell^p$ locally convex for $0<p<1$?
Where it is stated that $\ell^p(\mathbb{N})$ is not a locally convex space.
My Problem is that I do not understand the proof/answer. As far as I understand, I need to show that the topology (induced by the metric) can't come from a family of seminorms. (But I also do not know how to approach this.) Is there another way, to show that the space is not locally convex without using this definition?
Some help would be appreciated.
One helpful thing,was this post, which was kindly link by a user in the comments.