Let \(V\) be a vector space and \(V^*\) its dual space. We may characterize covariant tensors further by defining the tensor product more explicitly. Then for \(\omega, \eta\in V^*\) we define
\begin{equation*} \omega\otimes \eta(v_1,v_2):=\omega(v_1) \eta(v_2). \end{equation*}
Remark
- Of course we may generalize it for different vector space and higher number of them.
- Covariant tensor spaces may be identified with linear maps \(L(V_1,\ldots, V_n)\) (they are equivalent).