\psi(t)=\infty$. Consider the following data:

-   A family of Banach spaces $(X_h)_{h\in [0,1]}$ with $X_h\subseteq X_{h'}$ continuously for $h<h'$.

-   A mapping $T:X_1\to \mathcal{L}^{p+1}$ restricting to a continuous mapping $T_h:=T|_{X_h}:X_h\to \mathcal{L}^{p+h}$, for $h\in [0,1]$, such that there are measurable functions $$c_0,c_1:[0,1]\to [r,R],\quad \mbox{for some $0<r\leq R<\infty$},$$ with $$c_0(h)\|x\|_{X_h}\leq \|T_h(x)\|_{\mathcal{L}^{p+h}}\leq c_1(h)\|x\|_{X_h},\quad \forall h\in [0,1], \; x\in X_h.$$

Then $T$ defines a continuous mapping $T:X_\psi \to \mathcal{M}_\psi^{(p)}$ such that

1.  For any exponentiation invariant extended limit $\omega\in (L_\infty)^*$ $$\lim_{h\to  \tilde{\omega}} \frac{c_0(h)^{p}}{\psi(\mathrm{e}^{1/h})}\|x\|_{X_{h}}^{p+h}\leq \mathrm{Tr}_{\omega,\psi}(|T(x)|^p)\leq \lim_{h\to  \tilde{\omega}} \frac{c_1(h)^p}{\psi(\mathrm{e}^{1/h})}\|x\|_{X_{h}}^{p+h},$$ where $\tilde{\omega}$ is defined as in Equation . In particular, if $\lim_{h\to 0}\frac{c_0(h)}{c_1(h)}=1$, then $$\mathrm{Tr}_{\omega,\psi}(|T(x)|^p)= \lim_{h\to  \tilde{\omega}} \frac{c_0(h)^p}{\psi(\mathrm{e}^{1/h})}\|x\|_{X_{h}}^{p+h}.$$

2.  For any $x\in X_\psi$ we have that $$r\|x\|_{X_\psi}\leq \|T(x)\|_{\mathcal{M}_\psi^{(p)}}\leq R\|x\|_{X_\psi}.$$

3.  Assume that $\mathcal{N}$ is atomic. For any $x\in X_\psi$ we have that $$r\limsup_{h\searrow 0} \frac{1}{\psi(\mathrm{e}^{1/h})}\|x\|^{p+h}_{X_h}\leq \mathrm{d}_{\mathcal{M}_\psi^{(p)}}(|T(x)|^p,\mathcal{M}_{\psi,0})\leq \mathrm{e}R\limsup_{h\searrow 0} \frac{1}{\psi(\mathrm{e}^{1/h})}\|x\|^{p+h}_{X_h}.$$

In the setup of Corollary \[equivalencescor\], we note that the norms $\|x\|_{X_h}':=\|T(x)\|_{\mathcal{L}^{p+h}}$ on $X_h$ are equivalent to the norms $\|\cdot \|_{X_h}$. After this change of norms, we can take $c_0=c_1=1$ in which case Corollary \[equivalencescor\] is a trivial reformulation of Theorem \[allthethingsGSsaid\]. *The relevance of Corollary \[equivalencescor\] lies in that it is often possible to estimate the norms $\|x\|_{X_h}$ in situations where it is not possible to estimate 