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amilton
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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/
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/
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