Continuous seminorms

Question

If I have a locally convex vector space $S$ equipped with a countable family $(p_i)_{i \in I}$ of seminorms, is it correct that the topology remains unchanged if I add an extra seminorm $q$ satisfying : $\exists J \subset I$ ($J$ finite) $\exists c>0 | \forall x \in S$ $ q(x) \leq \Sigma_J p_i(x)$ ?

Answer 1

We start with the assumption that $S$ is equipped exactly with the topology induced by $(p_i)_{i\in I}$ - the smalled topology on $S$ which makes these seminorms continuous.

Since $q$ is a seminorm, it is continuous if and only if it is continuous at zero.

Take a sequence $x_n\to 0$. Then for all $j$, since $p_j$ are continuous, we have $p_j(x_n)\to p_j(0) = 0$ as $n\to \infty$.

Then, $$ 0\leq q(x_n) \leq \sum_{j\in J}p_j(x_n), $$

and using the sandwich theorem, we see that $q(x_n) \to 0 = q(0)$.

Consequently, adding $q$ in the original family of seminorms will not alter the topology.