/sci/ - Science & Math

I'm not sure what this question wants exactly. Is it something like, e.g. for (1, 1)-tensors [eqn]
\mathbf{T} = \left\{\begin{align}
& \mathbf{x} \in V_n \mapsto (\mathbf{\omega} \in V_n^* \mapsto T_j^i\ \mathbf{e}_i \otimes \mathbf{\theta}^j (\mathbf{\omega}, \mathbf{x})
\in \text{Hom}_\text{Vect}(V^*_n, \mathbb{R}) = V^{**}_n = V_n) \in \text{Hom}_\text{Vect}(V_n, V_n) \\
& \mathbf{\omega} \in V^*_n \mapsto (\mathbf{x} \in V_n \mapsto T_j^i\ \mathbf{e}_i \otimes \mathbf{\theta}^j (\mathbf{\omega}, \mathbf{x})
\in \text{Hom}_\text{Vect}(V_n, \mathbb{R}) = V^{*}_n) \in \text{Hom}_\text{Vect}(V^*_n, V^*_n) \\
\end{align}
\right. \\
\therefore \mathbf{T} \in \text{Hom}_\text{Vect}(V_n, V_n) \cup \text{Hom}_\text{Vect}(V^*_n, V^*_n)
[/eqn] and then I do something like that for all the other (r, s)?