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 $f(a)(t)=\exp(-\int_0^ta)$ as 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?

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?

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 the following: Given a smooth map $\chi : \mathbb{C}\rightarrow \mathbb{C}$ (that is well behaved, e.g. all derivatives are bounded, but also $\chi(z) = \exp(z)$ works) and a manifold $M$, concatenation $C^\infty(M)\rightarrow C^\infty(M), a \mapsto \chi \circ a$ is robust. This has some nice implications: E.g. if we define $f(a)(t)=\exp(-\int_0^ta)$, then $f$ is the coefficient-to-solution map for the initial value problem  $$\dot x +ax, \quad x(0)=1$$ and it is robust as a 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.

To summarise, my question is the following:

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

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 $f(a)(t)=\exp(-\int_0^ta)$ as 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?