Topology induced by seminorms and initial topology

Question

Let's say we have a family of seminorms $(\rho_\alpha)_{\alpha \in A}$ on a vector space $V$.

There are two ways to topologize $V$ using those seminorms:

1. We define topology $\mathcal S$ by a subbasis consisting of sets of the form $$ B_{x, \alpha, \epsilon} =\{v \in V : \rho_\alpha(v - x) < \epsilon \} $$ for some $\epsilon > 0, x \in V, \alpha \in A$.

2. We define topology $\mathcal I$ as the smallest topology in which the seminorms are continuous.

I've showed that the seminorms are continuous in $\mathcal S$, and that implies $\mathcal I \subseteq \mathcal S$. I'm having trouble with the other direction. It is clear to me that $B_{0, \alpha, \epsilon} = \rho_\alpha^{-1} ((-\epsilon, \epsilon ))$, so it's enough to show that vector addition is continuous in respect to $\mathcal I$, but I don't know how to do that.

Answer 1

Apart from trivial cases - all seminorms are identically $0$ - the two topologies don't coincide.

A seminorm cannot distinguish $x$ and $-x$, so every neighbourhood of $x$ in $\mathcal{I}$ contains $-x$. For an $x$ and a seminorm $p$ with $p(x) \neq 0$, there is however a ball $\{ x : p(y-x) < \epsilon\}$ that doesn't contain $-x$.

If you take the initial topology with respect to the family

$$\{ x \mapsto \rho_\alpha(x-y) : \alpha \in A, y \in V\},$$

then you describe $\mathcal{S}$ as an initial topology.