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)$ ?

  • $\begingroup$ you can better express your question, introducing some notation and definition? honestly I do not understand. $\endgroup$ – Andrew Oct 4 '16 at 21:57

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.