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?
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 $I \times \mathbb{R}^n$.