# Topologies generated by the two family of seminorms coincide

Suppose $P$ is a separating family of seminorms on a vector space $X.$ Let $L$ be the smallest family of seminorms on a vector space $X$ that contains $P$ and is closed under max. Let $\tau_P$ and $\tau_L$ denote the topologies on $X$ generated by $P$ and $L$ respectively. Show that these two topologies coincide.

My attempt:

I have shown that $\tau_L\subseteq \tau_P.$ To show the reverse inclusion:

Let $A \in \tau_P.$ Let $$B=\left\{\bigcap_{(p,n)\in I} V(p,n):I \subseteq P\times \mathbb{N}, I\text{ is finite}\right\}$$ where $$V(p,n)=\left\{x \in X:p(x)<\frac 1n\right\}$$ Then $B$ is a basis for $\tau_P$. Therefore, $$A=\bigcup_{x\in A\\B_x \in B} x+B_x$$

I'm unable to conclude from this that $A\in \tau_L.$ I especially don't understand how "closed under max" helps here.

Edit: As mentioned in the comments by Jochen, $P \subseteq L \implies \tau_P \subseteq \tau_L.$ Then I don't see the importance of $L$ being closed under max.

• It is clear that $\tau_L$ is finer than $\tau_P$ because $P\subseteq L$. Oct 23, 2017 at 6:48
• @Jochen, Then what is the importance of "closed under max." I haven't used it to prove $\tau_L \subseteq \tau_P.$ Oct 23, 2017 at 18:08
• For fixed center and radius the intersection of the balls w.r.t. some seminorms is the ball w.r.t. the maximum of the seminorms. Oct 23, 2017 at 20:17

Here I use slightly different symbols for those two sets of seminorms, which is from an exercise in Rudin's book. Let $$\mathscr{Q}$$ be the smallest family of seminorms on a vector space $$X$$ that contains $$\mathscr{P}$$ and is closed under max.
Firstly, by the usual technique in proving equality of two sets it's not hard to show: $$\mathscr{Q}=\{\max_{p \in P}p: P\text { a finite subset of } \mathscr{P}\}$$ Associate to each $$p \in \mathscr{P}$$ and to each positive integer $$n$$ the set $$V(p, n)=\left\{x: p(x)<\frac{1}{n}\right\}$$ Then $$\tau_{\mathscr{P}}$$ is the collection of all finite intersections of the sets $$V(p, n)$$.
Now we compare $$\tau_{\mathscr{P}}$$ and $$\tau_{\mathscr{Q}}$$.
On the one hand, $$\tau_{\mathscr{P}} \subset \tau_{\mathscr{Q}}$$ since $$\mathscr{P} \subset \mathscr{Q}$$. On the other hand, $$\forall q \in \mathscr{Q}$$, using the fact that $$q$$ is the max of finite number of seminorms in $$\mathscr{P}$$, we conculude: $$V(q, n)=\bigcap_{p \in P}\{x: p(x)<\frac{1}{n}\}$$ where $$P$$ is a finite subset of $$\mathscr{P}$$ such that $$q = \max_{p \in P}p$$. Thus we have shown $$\tau_{\mathscr{Q}} \subset \tau_{\mathscr{P}}$$.