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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.

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  • $\begingroup$ It is clear that $\tau_L$ is finer than $\tau_P$ because $P\subseteq L$. $\endgroup$
    – Jochen
    Oct 23, 2017 at 6:48
  • $\begingroup$ @Jochen, Then what is the importance of "closed under max." I haven't used it to prove $\tau_L \subseteq \tau_P.$ $\endgroup$ Oct 23, 2017 at 18:08
  • $\begingroup$ 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. $\endgroup$
    – Jochen
    Oct 23, 2017 at 20:17

1 Answer 1

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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}} $.

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