Why are there (at least) two definitions of a tensor? For some people a tensor is a product of vectors and covectors, but for others it's a functional. While it's true that the two points of view are equivalent (there's an isomorphism) I find having to switch between them confusing, as a beginner.

In a sense, the two points of view seem one the dual of the other because ##V## becomes ##V^*## and ##V^*## becomes ##V^{**}\sim V##, i.e. a tensor in ##V\otimes \cdots \otimes V \otimes V^*\otimes \cdots \otimes V^*## becomes a tensor in ##V^*\otimes \cdots \otimes V^* \otimes V\otimes \cdots \otimes V## seen as a functional which, basically, takes tensors of the first definition: $$

T(v_1, \ldots, v_p, w^1, \ldots, w^q) \sim

T(v_1\otimes\cdots\otimes v_p\otimes w^1\otimes\cdots\otimes w^q)

$$ Moreover, ##T(e_I\otimes (e^*)^J) = T_I^J## i.e. the functional gives the components for the tensor in the first definition.

If this is correct, shouldn't we really talk about

In a sense, the two points of view seem one the dual of the other because ##V## becomes ##V^*## and ##V^*## becomes ##V^{**}\sim V##, i.e. a tensor in ##V\otimes \cdots \otimes V \otimes V^*\otimes \cdots \otimes V^*## becomes a tensor in ##V^*\otimes \cdots \otimes V^* \otimes V\otimes \cdots \otimes V## seen as a functional which, basically, takes tensors of the first definition: $$

T(v_1, \ldots, v_p, w^1, \ldots, w^q) \sim

T(v_1\otimes\cdots\otimes v_p\otimes w^1\otimes\cdots\otimes w^q)

$$ Moreover, ##T(e_I\otimes (e^*)^J) = T_I^J## i.e. the functional gives the components for the tensor in the first definition.

If this is correct, shouldn't we really talk about

*cotensors*?
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