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There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\left(\frac{1}{n}e\right)\to 0$ and $\rho(te)\le \rho(e)$, for every $e\in E$ and $t\in [-1,1]$. It is well-known that any vector space topology is generated by a certain collection of F-semi-norms.

Can we always select a collection of incomparable F-semi-norms which generates the topology of a TVS?

By "incomparable" I mean that neither $\rho\le \lambda$ nor $\rho\ge \lambda$ hold.

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  • $\begingroup$ Why do you want to know this? Ususally, the opposite property of a directed generating set of F-norms is convenient. $\endgroup$
    – Jochen Wengenroth
    Commented Sep 20, 2023 at 7:53
  • $\begingroup$ @JochenWengenroth For a collection $P$ of F-seminorms on $E$ consider the following convergence. A net $(e_i)$ converges to $0$ if for every $\rho\in P$ and $\varepsilon>0$ there is $\lambda\in P$ with $\lambda\le \rho$ and $i_0$ such that $\lambda(e_i)<\varepsilon$, for $i\ge i_0$. I want to show that any linear topology on $E$ can be described this way. $\endgroup$
    – erz
    Commented Sep 20, 2023 at 13:39

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