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In Pierre-Louis Lions' famous work Axioms and fundamental equations of image processing, he defines $C_b^\infty$ on Page 9 as

...on $C_b^\infty$, i.e., the space of bounded functions having bounded derivatives at any order.

And on the next page, he defines $Q$, a subset of $C_{b}^{\infty}$

$$ Q=\left\{f \text { in } C_{b}^{\infty}, \forall n \geq 0,\left\|D^{\alpha} f\right\|_{\infty} \leq C_{n} \text { for all }|\alpha|=n\right\} $$ and $C_n$ is an arbitrary increasing sequence of nonnegative constants.

What is the difference between $Q$ and $C_{b}^{\infty}$? For me they should be identical. Am I missing something?

Wang
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  • $$C^\infty_b={ f\in C^\infty,:,\exists {C_n}{n\in\Bbb N},\forall n\ge 0,\forall \lvert \alpha\rvert=n,\lVert D^\alpha f\rVert\infty\le C_n}$$ whereas, given some $C:={C_n}{n\in\Bbb N}$ (such that yada yada yada), $$Q=Q_C={f\in C_b^\infty,:, \forall n\ge 0,\forall \lvert \alpha\rvert=n,\lVert D^\alpha f\rVert\infty\le C_n}$$ – Sassatelli Giulio Aug 05 '22 at 09:40
  • Sounds pretty reasonable. Anyone has other idea? – Wang Aug 06 '22 at 01:29

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