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Disclaimer: I'm new to general topology, so please bear with me. I've got a strong background in measure theory and the following seems to be complete analogous to $\sigma$-algebras generated by systems of sets and $\sigma$-algebras generated by families of functions into given measurable spaces.

I'm not sure if I understand this correctly, but if I do, then "the locally convex topology $\tau$ generated by a family $(p_i)_{i\in I}$ (where $I$ is a nonempty set) on a $\mathbb R$-vector space $E$", is precisely given by$^1$ $$\tau:=\tau(p_i,i\in I).\tag1$$

Given that, I would like to show that any seminorm $q$ on $E$ is $\tau$-continuous iff $$q(x)\le c\max_{J\subseteq I}p(x)\;\;\;\text{for all }x\in E\tag2$$ for some $c\ge0$ and some finite $J\subseteq I$.

It's clear to me that if $\tau'$ is any topology on $E$ such that $(E,\tau')$ is a topological $\mathbb R$-vector space, then

  1. $q$ is $\tau'$-continuous;
  2. $q$ is $\tau'$-continuous at $0$;
  3. $\{x\in E:q(x)<1\}\subseteq\mathcal N_{\tau'}(0)$

are equivalent, where $$\mathcal N_{\tau'}(x):=\left\{N\subseteq E:N\text{ is a }\tau'\text{-neighborhood of }x\right\}\;\;\;\text{for }x\in E$$ and we've clearly got $$\mathcal N_{\tau'}(x)=x+\mathcal N_{\tau'}(0)\;\;\;\text{for all }x\in E\tag3$$ and $$\mathcal N_{\tau'}(\lambda x)=\lambda\mathcal N_{\tau'}(x)\;\;\;\text{for all }\lambda\in\mathbb R\setminus\{0\}\text{ and }x\in E.\tag4$$

However, I really struggle to show the desired claim. So, how can we show that?

$^1$ Since I haven't found this in any reference, I've made the following up by myself, but I guess it should be correct:

(a) If $\mathcal E\subseteq 2^E$, then $$\tau(\mathcal E):=\bigcap_{\substack{\tau\text{ is a topology on }E\\\mathcal E\:\subseteq\:\tau}}$$ is the smallest toplogy on $E$ containing $\mathcal E$.

(b) If $(E',\tau')$ is a topological space and $f:E\to E'$, then $$\tau(f):=\left\{f^{-1}(\Omega'):\Omega'\in\tau'\right\}$$ is the smallest topology $\tau$ on $E$ such that $f$ is $(\tau,\tau')$-continuous.

(c) If $(E_i,\tau_i)$ is a topological space and $f_i:E\to E_i$ for $i\in I$, then $$\tau(f_i,i\in I):=\tau\left(\bigcup_{i\in I}\tau(f_i)\right)$$ is the smallest topology $\tau$ on $E$ such that $f_i$ is $(\tau,\tau')$-continuous for all $i\in I$.

This is completely analogous to what we are doing in measure theory with "topology" replaced by $\sigma$-algebra and "continuous" replaced by "measurable".

0xbadf00d
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  • I would have to think about it to be sure, but what you say in the footnote c) is not how you usually define the locally convex topology from a set of seminorms. A definition which works is explained in https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam Dec 07 '20 at 15:42
  • @AbdelmalekAbdesselam The construction of the locally convex topology in terms of neighborhoods should be equivalent. The locally convex topology should be the initial topology making all seminorms continuous: https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space#Seminorm_topology. – 0xbadf00d Dec 07 '20 at 20:10
  • I'm sure my definition is correct but I am not sure the passage from wikipedia is correct and don't have time to find the flaw if indeed there is one. If $p$ is one of your seminorms and $T$ makes it continuous then sets like ${y\ |\ p(y)<\epsilon}$ must be open, i.e., in $T$. But what you need are sets ${y\ |\ p(y-x)<\epsilon}$, for all $x$'s. You could try to argue $p$ is continuous for $T$ and $y\mapsto y-x$ is continuous as part of the TVS axioms, and so you just compose continuous functions. The problem is you don't yet have a TVS structure. – Abdelmalek Abdesselam Dec 07 '20 at 20:51
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    Okay, Wikipedia if officially wrong. See https://math.stackexchange.com/questions/1284692/topology-induced-by-seminorms-and-initial-topology – Abdelmalek Abdesselam Dec 07 '20 at 20:56
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    What @AbdelmalekAbdesselam said. Nevertheless, if $\tau$ denotes the topology induced by the family $(p_i)$ of seminorms (the correct one), then $(2)$ characterises the $\tau$-continuous seminorms. – Daniel Fischer Dec 07 '20 at 21:12

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