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 satisfies 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?

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?