How to reconcile these two tensor notations?

Question

How can I connect the symbols, i.e. the notation, preferably in English or with a 2 x 2 (or 3 x 3) matrix example, between an order 2 tensor expressed as:

A $(p,q)$ tensor, $T$ is a MULTILINEAR MAP that takes $p$ copies of $V^*$ and $q$ copies of $V$ and maps multilinearly (linear in each entry) to $k:$

$$T: \underset{p}{\underbrace{V^*\times \cdots \times V^*}}\times \underset{q}{\underbrace{V\times \cdots \times V\times V}} \overset{\sim}\rightarrow K\tag 1$$

and

$$\large \mathbf{T}= T_{ij}\;\mathbf{\hat e_i}\otimes\mathbf{\hat e_j}\tag 2$$

?

Would $\large T_{ij}$ in Eq.2, which I guess can be interpreted as coefficients or entries in a matrix) be the $V^*$ elements of the dual space (functionals), while the $\mathbf{\hat e_i}$ and $\mathbf{\hat e_j}$ are the vectors $V$? Are the $\times$ in Eq.1 Cartesian products (presumably they can't be cross-products...)? Are the $V$'s in Eq. 1 just vectors, or are they elements of the double dual? Are the indices $(p,q)$ in Eq.1 the equivalent of $(i,j)$ in Eq.2?

I realize Eq. 1 is probably more general, but it should be possible to reduce it to the more simple case of Eq.2, again just to be able to enunciate what the symbols are. Do both equations produce a field element?

Answer 1

The $\times$ in eq. 1 are Cartesian products. Note that, in finite dimensions, $V^{**}=V$, so vectors can be seen as elements of the double dual.

Now, if $\mathcal T^{(p,q)}(V)$ denotes the space of $(p,q)$ tensors over a vector space $V$, $\mathcal T^{(p,q)}(V)$ is a vector space, and if we pick $\{e_1, \cdots, e_n\}$ a basis for $V$, and $\{\omega_1, \cdots, \omega_n\}$ the dual basis, we can construct a basis for $\mathcal T^{(p,q)}(V)$ with the elements:

$$e_{i_1} \otimes \cdots \otimes e_{i_p} \otimes \omega^{j_1} \otimes \cdots \otimes \omega^{j_q}$$

with $\{i_1, ..., i_p, j_1, ..., j_q\}\in\{1,...,n\}$.

So if $T\in \mathcal T^{(p,q)}(V)$, then it would be \begin{equation} T=\sum \lambda_{i_1, \cdots, i_p}^{j_1, \cdots, j_q} e_{i_1} \otimes \cdots \otimes e_{i_p} \otimes \omega^{j_1} \otimes \cdots \otimes \omega^{j_q} \end{equation}

Equation 2 is the particular case of the previous equation for $\mathcal T^{(0,2)}(V)$ (we will write, instead of $i_1,i_2$, $i,j$):

$$T=\lambda_{ij}\ \omega^i \otimes \omega^j$$

As you only have two indices, you can form a matrix with the numbers $(\lambda_{ij})$.