The tensor space \(T^{(k,l)}(V)\) of some finite dimensional vector space is isomorph to the space of multilinear maps

\begin{equation*} \underbrace{V^*\times \cdots \times V^*}_\text{\(k\)-times} \times \underbrace{V\times \cdots \times V}_\text{\(l\)-times} \to \mathbb{R} . \end{equation*}

Remark

We have also such an isomorphism with \(V\) as the codomain. The tensor space \(T^{(k+1,l)}(V)\) is isomorph to the space of multilinear maps \begin{equation*} \underbrace{V^*\times \cdots \times V^*}_\text{\(k\)-times} \times \underbrace{V\times \cdots \times V}_\text{\(l\)-times} \to V . \end{equation*} [@lee2018riemannian_manifolds, Proposition B.1]

Links Link to heading

\(\mathfrak{X}(M)\) as a codomain of a smooth tensor field

Questions Link to heading

Are other codomains possible?