In my current work the following property of maps between locally convex spaces showed up at several places and proved to be useful. It seems quite elementary to me, so I would like to know whether it already has an established name or is equivalent to some other property.

Let $f: U\rightarrow F$ be a (possibly non-linear) map between an open set $U\subset E$ of a locally convex space into another locally convex space $F$. Suppose for every continuous semi-norm $\Vert \cdot \Vert$ on $F$ there exists a non-decreasing function $\omega:[0,\infty)\rightarrow [ 0,\infty)$ and a continuous semi-norm $\Vert \cdot \Vert'$ on $E$ such that $$ \Vert f(x)\Vert \le \omega(\Vert x \Vert')\quad \text{ for all } x \in U.$$ Let's for now call such a map 'robust'. It's clear that the concatenation of robust maps, when defined, is robust again. Further, linear maps (defined on $U=E$) are robust if and only if they are continuous.

My main example of a non-linear robust map is concatenation $$ C^\infty(M)\rightarrow C^\infty(M), a \mapsto \chi \circ a, $$ where $M$ is a smooth manifold and $\chi:\mathbb{C}\rightarrow \mathbb{C}$ is a smooth function that is well behaved, e.g. all of its derivatives are bounded, but also $\chi(z)=\exp(z)$ works. This has some nice implications: For example the parameter-to-solution map for the initial value problem $$\dot x +ax, \quad x(0)=1$$ can be written as $f(a)(t)=\exp(-\int_0^ta)$ and is thus easily seen to define a robust map $f:C^\infty[0,\infty)\rightarrow C^\infty[0,\infty)$. A more sophisticated consequence is that one can construct parametrices of classical pseudodifferential operators in a robust way (concatenate the principal symbol with $\chi(z)=z^{-1}$ (smoothed near $0$)).

In general robustness seems to come up as a property of 'parameter dependence' in a smooth setting. Typically in this setting one already has a continuous dependency, which allows to make constructions, constants etc. uniform in small neighbourhoods of a fixed parameter. Robustness allows to get uniformity over large classes of parameters, as long as an appropriate semi-norm ($\Vert \cdot \Vert'$ above) stays bounded. This is relevant for recovering parameters in statistical inverse problems.

To summarise, my question is the following:

Question. Is robustness equivalent or related to some well known property in functional analysis or has it been studied anywhere?