**Definition** Let $E\_{1} \xrightarrow{\pi\_{1}} M\_{1}$, $E\_{2} \xrightarrow{\pi\_{2}} M\_{2}$ be two vector bundles over $M\_{1}$ and $M\_{2}$ with fibers $V\_{1}$, $V\_{2}$ respectively. The exterior tensor product $E\_{1} \boxtimes E\_{2}$ is defined as the vector bundle over $M\_{1} \times M\_{2}$, whose fiber over $(x, y) \in M\_{1} \times M\_{2}$ is $E\_{1 x} \otimes E\_{2 y}$, where $E\_{1 x}$ is the fibre of $E\_{1}$ over $x$ and $E\_{2 y}$ is the fibre of $E\_{2}$ over $y$.


In the article [Brouder, Dang, Laurent-Gengoux, and Rejzner - Properties of field functionals and
characterization of local functionals](https://hal.archives-ouvertes.fr/hal-01654120) they have the so called fundamental result on the projective tensor product of sections of a vector bundle



> 
> Proposition III.8. Let $\Gamma(M, B)$ be the space of smooth sections
> of some smooth finite rank vector bundle $B \rightarrow M$ on a
> manifold $M$. Then $\Gamma(M, B)^{\hat{\otimes}\_{\pi}
>  k}=\Gamma\left(M^{k}, B^{\boxtimes k}\right)$.
> 
> 
> 


Where $\boxtimes$ means exterior tensor product and $\otimes\_\pi$ is the projective tensor product.


I am trying to prove this theorem. My first step is to prove the algebraic isomorphism.
This is how I am trying to prove the algebraic isomorphism by following the same steps as in [A nonlinear theory of generalized tensor fields
on Riemannian manifolds](https://core.ac.uk/download/pdf/11592565.pdf) by Eduard Nigsch at page 89:


Let $V$ be the fiber of the **trivial vector bundle** $B \xrightarrow{\pi} M$.
We know there is a bilinear map from $\alpha :V\times V \rightarrow V\otimes V$ with $$\alpha(v\_1,v\_2)=v\_1 \otimes v\_2$$


where $v\_1,v\_2 \in V$.


Now let $\psi :B \rightarrow U\times V$ be a local trivialization map with $U \subset M$.


Now we define the local sections $\sigma\_i$ by
$\sigma\_i(x) =\psi^{-1}(x,v\_i)$.


We also define the local sections $\gamma\_{ij}$ by
$\gamma\_{ij} =\phi^{-1}(x,y,v\_i\otimes v\_j)$
where $\phi :B{\boxtimes}B \rightarrow U^2 \times V\otimes V$ is a local trivialization map of the bundle $B{\boxtimes}B \rightarrow M \times M$.


Define the map
$g :\Gamma\left(M, B\right) \times \Gamma\left(M, B\right) \rightarrow \Gamma\left(M\times M, B{\boxtimes}B\right)$ by
$$g=\sum\_{i, j} m \circ\left(i d \times m\left(\cdot, \gamma\_{i j}\right)\right) \circ\left(\sigma\_{i}^{\*} \times \sigma\_{j}^{\*}\right)$$


where $m: C^{\infty}(M) \times \Gamma(B\otimes B)\rightarrow \Gamma(B\otimes B) $ is the module multiplication on $\Gamma(B\otimes B)$ and $\sigma\_{i}^{\*}$ is the dual of $\sigma\_{i}$.


By the properties of the tensor product there is a bilinear map $f: \Gamma(M, B)\otimes \Gamma(M, B)\rightarrow \Gamma\left(M\times M, B{\boxtimes}B\right)$ such that if $g$ is the map $h:\Gamma\left(M, B\right) \times \Gamma\left(M, B\right) \rightarrow \Gamma(M, B)\otimes \Gamma(M, B)$
we have that $$g=f \circ h.$$


For the inverse define $h^{-1}: \Gamma\left(M\times M, B{\boxtimes}B\right)\rightarrow \Gamma(M, B)\otimes \Gamma(M, B)$
$h^{-1}(s)=\sum\_{i, j} \gamma\_{i j}^{\*}(s) \alpha\_{i} \otimes \beta\_{j}$ for $s \in \Gamma\left(M\times M, B{\boxtimes}B\right)$


Now we have that $$
\begin{aligned}
&h^{-1}\left(f\left(t \otimes u\right)\right)=h^{-1}\left(h \left(t^{i} \alpha\_{i}, u^{j} \beta\_{j}\right)\right)=h^{-1} \left(t^{i} u^{j} \gamma\_{i j}\right)=t^{i} u^{j} \alpha\_{i} \otimes \beta\_{j}\\
&=t^{i} \alpha\_{i} \otimes u^{j} \beta\_{j}=t \otimes u \text { and }\\
&h(h^{-1}(s))=h\left(s^{i j} \alpha\_{i} \otimes \beta\_{j}\right)=s^{i j} g\left(\alpha\_{i}, \beta\_{j}\right)=s^{i j} \gamma\_{i j}=s .
\end{aligned}
$$


Is my proof correct?



Fundamental result on the projective tensor product of sections of a vector bundle
The tensor product (either one) is symmetric $A \otimes B \cong B \otimes A$, associative $(A\otimes B) \otimes C \cong A \otimes (B \otimes C)$ and distributive over direct sums $A \otimes (B \oplus C) \cong (A\otimes B) \oplus (A\otimes C)$. Spaces of sections preserve direct sum decompositions $\Gamma(M, E\_1 \oplus E\_2) \cong \Gamma(M,E\_1) \oplus \Gamma(M,E\_2)$. Any vector bundle $E\to M$ can be realized as a summand of a trivial vector bundle $(V\times M \to M) \cong (E\oplus E' \to M)$, where we can realize the sub-bundles $E$ and $E'$ and the image and kernel of a fiber-wise projection $P$ on $V\times M$ ([it is sufficient](https://en.wikipedia.org/wiki/Vector_bundle#Operations_on_vector_bundles) that $M$ is contractible to a compact space). The completed tensor product satisfies the identity $C^\infty(M\_1) \otimes C^\infty(M\_2) \cong C^\infty(M\_1 \times M\_2)$ (for details see Trèves *opological Vector Spaces, Distributions and Kernels* 1970).


With the above background, the proof is straightforward. Start with the observation that for a trivial bundle $\Gamma(M, V\times M) \cong C^\infty(M) \otimes V$. Hence $$\Gamma(M\_1,V\_1\times M\_1) \otimes \Gamma(M\_2,V\_2\times M\_2)\cong C^\infty(M\_1\times M\_2) \otimes (V\_1\otimes V\_2) \cong \Gamma(M\_1\times M\_2, (V\_1 \times M\_1) \boxtimes (V\_2 \times M\_2)).$$
Applying direct sum decompositions to both sides gives
$$
 (\Gamma(M\_1,E\_1) \oplus \Gamma(M\_1,E'\_1)) \otimes (\Gamma(M\_2,E\_2) \oplus \Gamma(M\_2,E'\_2))
 \cong \Gamma(M\_1\times M\_2, (E\_1\oplus E'\_1) \boxtimes (E\_2\oplus E'\_2)) .
$$
Finally, the desired isomorphism
$$
 \Gamma(M\_1,E\_1) \otimes \Gamma(M\_2,E\_2)
 \cong \Gamma(M\_1\times M\_2, E\_1 \boxtimes E\_2)
$$
follows because both sides coincide with the image of the projection $P\_1\otimes P\_2$ (interpreted as acting fiber-wise or on spaces of sections as needed).


