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Let $(F,d)$ be a Fréchet space over $\mathbb{F}\in \{\mathbb{C},\mathbb{R}\}$. Are there conditions under which, there exists some $C,d>0$ such that: for every $f\in F$ and every $k\in \mathbb{F}$ it holds that $$ d(kf,0)\leq C |k|^d \,d(f,0). $$

Clearly, $C=p=1$ is satisfied if and only if $(F,d)$ is Banach; i.e. $d$ is a norm.However, what about the general case?

Namely, is there a characterization/description of such spaces known in the literature?

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    $\begingroup$ For $C=1$ and $d=p\in (0,1]$ such spaces are called $p$-normed topological vector spaces. Some information is in Köthe's book Topological Vector Spaces 1. $\endgroup$
    – Jochen Wengenroth
    Commented Jun 28, 2022 at 7:55
  • $\begingroup$ @JochenWengenroth Ah, okay; I'll take a look. Do you know of a typical concrete example of such a space? $\endgroup$
    – ABIM
    Commented Jun 28, 2022 at 15:01
  • $\begingroup$ The main examples are $L_p(\mu)$, $0<p<1$. $\endgroup$
    – Bill Johnson
    Commented Jul 14, 2022 at 15:30
  • $\begingroup$ @BillJohnson Fair enough (These are the only ones I generated myself also). $\endgroup$
    – ABIM
    Commented Jul 15, 2022 at 6:54

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