Let $X$ be Hausdorff and let $\mathcal{P}$ be a (countable) family of seminorms on $X$ which generate a topology on $X$ via the neighbourhoods $B_{p}(x,r):=\{x\in X:p(x)<r\}$, for $p\in\mathcal{P}$ and $r>0$. Now, define a new family of seminorms $\widetilde{\mathcal{P}}$ on $X$ of the form $\widetilde{p}:=\max_{1\le j\le k}p_j$ for $k\in\mathbb{N}$ and $p_j\in\mathcal{P}$.

I want to show that the balls $B_{\widetilde{p}}(x,r_{\widetilde{p}})$ form a local base in $X$. However, I am struggling to prove this, even from an intuitive perspective.

Essentially, we want to show that if $U$ is any neighbourhood of $0$ in $X$, then there exists a $\widetilde{p}\in\widetilde{\mathcal{P}}$ such that $B_{\widetilde{p}}(0,r)\subset U$.

However, due to the construction of our topology, $U$ is of the form $B_p(0,r_p)$ for $p\in\mathcal{P}$ and $r_p>0$. Hence we want to show, in fact, that for all $p_j\in\mathcal{P}$ and $r_{j}>0$, there exists a $\widetilde{p}\in\widetilde{\mathcal{P}}$ and $r_{\widetilde{p}}>0$ such that $B_{\widetilde{p}}(0,r_{\widetilde{p}})\subset B_{p_j}(0,r_j)$ for all $j\in[1,k]$.

But this seems counter intuitive, since for all $p_1,...,p_k\in\mathcal{P}$ and $r_1,...,r_k>0$, there exists $\widetilde{p}\in\widetilde{\mathcal{P}}$ such that $\max\{p_1(x),...,p_k(x)\}=\widetilde{p}(x)<r_{\widetilde{p}}$, which implies that $r_{\widetilde{p}}>r_{j}$ for all $p_j\in\mathcal{P}$ (with $0\le j\le k)$ and therefore the "containment" would be the reverse of what I want to actually prove.


The intuition is that inequalities get reversed at key moments.

Write $\tilde p_k = \max_{1\leq j \leq k} p_j$. Then $p_k(x) \leq \tilde p_k(x)$, but the reverse inequality is true for balls: $B_{\tilde p_k}(0,r) \subseteq B_{p_k}(0,r)$. In other words, the bigger the seminorm, the smaller the corresponding balls.

That the seminorms $\tilde p_k$ form a local base now follows from the inclusions $$ \bigcap_{1\leq j \leq k} B_{p_j}(x,r) \subseteq B_{\tilde p_k}(x,r) \subseteq B_{p_k}(x,r)\,,$$ which are valid for all $x \in X$ and $r > 0$.


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.