I'm trying to understand the concepts of matricization (matrix unfolding) and vectorization of tensor products. In the past, I've only dealt with tensor products of infinite-dimensional Banach and Hilbert spaces and hence my my view on tensor products differs from most sources introducing the aforementioned concepts. So, I would be really thankful if someone could explain to me how they fit into my understanding.
If $E_i$ is a $\mathbb R$-vector space, I'm defining $$(x_1\otimes x_2)(B):=B(x_1,x_2)\;\;\;\text{for }B\in\mathcal B(E_1\times E_2)\text{ and }x_i\in E_i,$$ where $\mathcal B(E_1\times E_2)$ is the space of bilinear forms on $E_1\times E_2$, and $$E_1\otimes E_2:=\operatorname{span}\{x_1\otimes x_2:E_i\in E_i\}\subseteq{\mathcal B(E_1\times E_2)}^\ast.$$ If $B_i$ is a basis of $E_i$, then $\{e_1\otimes e_2:e_i\in B_i\}$ is a basis of $E_1\otimes E_2$ and hence, if $\dim E_i\in\mathbb N$, then $\dim(E_1\otimes E_2)=\dim E_1\dim E_2$.
If $F_i$ is a $\mathbb R$-vector space and $A_i:E_i\to F_i$ is linear, the linearization $A_1\otimes A_2$ of $$E_1\times E_2\ni(x_1,x_2)\mapsto A_1x_1\otimes A_2x_2\tag1$$ is a linear operator from $E_1\otimes E_2$ to $F_1\otimes F_2$.
$E_1^\ast\otimes E_2$ is naturally embedded into $\mathcal L(E_1,E_2)$, $$(\varphi\otimes y)(x)=\varphi(x)y\;\;\;\text{for all }(\varphi,y)\in E_1^\ast\times Y.\tag2$$
If $H_i$ is a pre-$\mathbb R$-Hilbert space, there is a unique inner product $\langle\;\cdot\;,\;\cdot\;\rangle_{H_1\otimes H_2}$ on $H_1\otimes H_2$ with $$\langle x_1\otimes y_1,x_2\otimes y_2\rangle_{H_1\otimes H_2}=\langle x_1,x_2\rangle_{H_1}\langle y_1,y_2\rangle_{H_2}\tag3$$ for all $(x_1,y_1),(x_2,y_2)\in H_1\times H_2$.
Now, if $d_i:=\dim H_i\in\mathbb N$, it is clear to me that we may fix orthonormal bases $(e_1,\ldots,e_{d_1})$ and $(f_1,\ldots,f_{d_2})$ of $H_1$ and $H_2$, respectively, and denote \begin{align}x_j&:=\langle x,e_j\rangle_{H_1},\\y_k&:=\langle y,f_k\rangle_{H_2}\end{align} for $x\in E_1,y\in E_2$ and $j\in\{1,\ldots,d_1\},k\in\{1,\ldots,d_2\}$. Now, clearly, $H_i\cong\mathbb R^{d_i}$ and I guess it's assumed that $H_i=\mathbb R^{d_i}$ and $(e_1,\ldots,e_{d_1})$ and $(f_1,\ldots,f_{d_2})$ are the standard bases of $\mathbb R^{d_1}$ and $\mathbb R^{d_2}$, respectively.
How precisely are matricization and vectorization now defined? I'm really trying to understand how these things fit into the more abstract setting described before.
Remark: It's clear to me that if $A\in\mathbb R^{d_2\times d_1}$, then we may treat $A$ as a linear operator from $\mathbb R^{d_1}$ to $\mathbb R^{d_2}$ which is the one identified with $$\sum_{i=1}^{d_1}e_i\otimes Ae_i\tag4$$ (noting that $Ae_i$ is the $i$th column of $A$.)
EDIT: If the linear operators $A_i:H_i\to F_i$ can be identified with \begin{align}A_1&=\sum_{i=1}^{d_1}e_i\otimes u_i,\\A_2&=\sum_{i=1}^{d_2}f_i\otimes v_i\tag5\end{align} for some $u_i,v_i$ and $x\in H_1,y\in H_2$, then $$(A_1\otimes A_2)(x\otimes y)=A_1(x)\otimes A_2(y)=\sum_{i=1}^{d_1}\langle x,e_i\rangle_{H_1}u_i\otimes\sum_{i=1}^{d_2}\langle y,f_i\rangle_{H_2}v_i\tag6.$$ Now we can identify $x,y$ with $$\begin{pmatrix}\langle x,e_1\rangle_{H_1}\\\vdots\\\langle x,e_{d_1}\rangle_{H_1}\end{pmatrix},\begin{pmatrix}\langle x,e_1\rangle_{H_2}\\\vdots\\\langle x,e_{d_1}\rangle_{H_2}\tag7\end{pmatrix}$$ and I think we can identify $A_1\otimes A_2$ with some kind of matrix. And maybe this can be generalized to the completion of the tensor products with respect to the projective norm (the operators $A_1$ and $A_2$ correspond to nuclear / trace class operators then).