Interpretations and Representations of Classical Tensors

Abstract: Classical tensors, the familiar mathematical objects denoted by symbols such as $t_{i}$, $t^{ij}$ and $t_{k}^{ij}$, are usually interpreted either as 'coordinatizable objects' with coordinates changing in a specific way under a change of coordinate system or as elements of tensor spaces of the form $V^{\otimes n}\otimes\left(V^{*}\right)^{\otimes m}$. An alternative interpretation of classical tensors as linear tensor maps of the form $V^{\otimes m}\rightarrow V^{\otimes n}$ is presented here. In this interpretation, tensor multiplication is seen as generalized function composition. Representations of classical tensors by means of arrays are also considered.