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Is there a standard term for the quasi-norm $$\|f\|_{[k]}=\sum_{i=1}^k(\sup\|f^{(i)}\|)^{1/i}$$ ?

It is useful due to the fact that it is reasonably compatible with post-composition by smooth functions: $$\|F(f)\|_{[k]}\leq\mathrm{const}_{F,k}\cdot\|f\|_{[k]}$$ in contrast to the usual $C^k$ norm for which this inequality does not hold.

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    $\begingroup$ This seems similar to the definition of Gevrey classes? $\endgroup$
    – Willie Wong
    Commented Dec 1, 2022 at 1:48

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