Duistermaat and Kolk: definition of separating family of seminorms

Question

I am reading the book "Distributions: Theory and Applications" by Duistermaat and Kolk. In chapter 8, Definition 8.5., they say that a family of semi-norms $\mathcal{N}$ is separating if

1. For every $x\neq 0$, there exists an $n\in \mathcal{N}$ such that $n(x)\neq 0$.

2. For every $n,m \in \mathcal{N}$, there exists a $p\in \mathcal{N}$, such that $n(x),m(x)\le p(x)$, for all $x$.

However, in every other source I looked at, only the first requirement was stated. Do the authors include the second requirement because given a family of semi-norms that satisfies only the first, we can always extend it to one that also satisfies the second, by say including all finite sums?

Answer 1

Yes, you've got it, exactly. Probably for some technical reason they want to have the collection of seminorms be closed under positive-real linear combinations, etc.