Graduate Problem in Understanding notation of distributional section

[amilton]

In this post [Observables][1] By Urs Schreiber he denotes the space of distributional sections in defenition 7.9 by $\Gamma_{\Sigma}^{\prime}\left(E^*\right)$

That is if $u \in \Gamma_{\Sigma}^{\prime}\left(E^*\right)$ than $u$ is a linear functional that takes as argument sections of a vector bundle $E$

In the same post he has proposition 7.10

> Let $E \stackrel{f b}{\rightarrow} \Sigma$ be a smooth vector bundle over Minkowski spacetime and let $s \in\{c p, \pm c p, s c p, t c p\}$ be any of the support conditions from def. 2.36.
Then the operation of regarding a compactly supported smooth section of the dual vector bundle as a functional on sections with this support property is a dense subspace inclusion into the topological vector space of distributional sections from def. 7.9:
$$
\begin{array}{ccc}
\Gamma_{\Sigma, \mathrm{cp}}\left(E^*\right) & \stackrel{u_{(-)}}{\longrightarrow} & \Gamma_{\Sigma, S}^{\prime}(E) \\
b & \mapsto & \left(\Phi \mapsto \int_{\Sigma} b(x) \cdot \Phi(x) \operatorname{dvol}_{\Sigma}(x)\right)
\end{array}
$$

In my understanding $u_{()}$ is a map from the space of sections of the dual bundle to the space of the distributional section .Why $u_{()} \in \Gamma_{\Sigma, s}^{\prime}(E)$ ? Shouldn't we have $u_{()} \in \Gamma_{\Sigma, s}^{\prime}(E^*)$ [1]: https://www.physicsforums.com/insights/newideaofquantumfieldtheory-observables/

 

[member38644]

It is correct that in proposition 7.10, $u_{()}$ is a map from the space of sections of the dual bundle $E^*$ to the space of distributional sections $\Gamma_{\Sigma, s}^{\prime}(E)$. This can be seen from the notation $\Gamma_{\Sigma, s}^{\prime}(E)$ which indicates that we are considering distributional sections of $E$, not $E^*$.

The notation may be confusing because in definition 7.9, $\Gamma_{\Sigma}^{\prime}\left(E^*\right)$ is used to denote the space of distributional sections of $E^*$. However, in this case, the notation is being used to emphasize that the distributional sections are taken with respect to the dual bundle.

Therefore, in proposition 7.10, we are considering distributional sections of the original bundle $E$, not its dual $E^*$. This is why $u_{()}$ is an element of $\Gamma_{\Sigma, s}^{\prime}(E)$, not $\Gamma_{\Sigma, s}^{\prime}(E^*)$.