# Reading a french paper: What does "L'application est bornée de H dans X" mean

In a paper by Ginibre "Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace" he utilizes phrases like

"des estimations linéaires assurant que l’application $$\phi \rightarrow U(\cdot) \phi$$ est bornée de $$\mathcal{H}$$ dans $$X$$"

(here $$\mathcal{H}$$ is Hilbert and $$X$$ is Banach) a lot. Can someone tell me what this exactly means? More precisely I'm having trouble with understanding "bornée de $$\mathcal{H}$$ dans $$X$$", so literally "bounded from $$\mathcal{H}$$ in $$X$$". Maybe it means that $$\mathcal{H} \subset X$$? Or something completely different?

• It certainly means 2 things grouped in a too compact way : it is a mapping from $H$ to $X$ and its values are bounded, (i.e., $\phi(H)$ is bounded) Nov 11, 2019 at 12:36
• It means this. Nov 11, 2019 at 12:39
• @conditionalMethod Thanks, I know what a bounded operator is, that was unfortunately not the question :) Nov 11, 2019 at 13:07
• @JeanMarie Ok that also makes sense in the context. So to be completely sure: $\phi \rightarrow U(\cdot) \phi$ is a mapping from $H$ to $X$ and bounded as a mapping? Follow-up question: What do you mean by $\phi(H)$ being bounded in $X$? Do you mean that $\sup_{x\in \phi(H)} \| x \|_X < \infty$? Nov 11, 2019 at 13:11
• Take a look at the inequalities (2.1), (2.2), (3.4), (3.5), (3.10), (3.11), (3.12), (3.20), (3.21), ... Nov 11, 2019 at 13:16

I'm not much of a French speaker, but I'd read this as "the map $$\phi \mapsto U(\cdot)\phi$$ is bounded from $$\mathcal{H}$$ to $$X$$." In other words, considering the context, the linear operator $$\phi \mapsto U(\cdot) \phi$$, which maps $$\mathcal{H}$$ into $$X$$, is a bounded operator with respect to the norms on $$\mathcal{H}$$ and $$X$$; we have $$\|U(\cdot) \phi\|_X \le C \|\phi\|_{\mathcal{H}}$$ for some constant $$C$$.
Note that it seems $$\mathcal{H}$$ is understood to be some space of functions on some set such as $$\mathbb{R}^n$$, and $$U(\cdot)$$ is a one-parameter family of linear operators, parametrized by some interval $$I$$, so $$U(\cdot) \phi$$ means the function $$I \times \mathbb{R}^n \to \mathbb{R}$$ defined by $$(t,x) \mapsto (U(t)\phi)(x)$$. Thus $$X$$ should be some space of functions on $$I \times \mathbb{R}^n$$.