Is tensor always a scalar valued multi linear map?

Question

The definition of a Tensor as a multilinear map is usually given as

$$T:V^* \times V^* \times\ ... \times\ V \times V \to R$$

Where $V$ is the vector space and $V^*$ is its dual space and $R$ is the real field . Is tensor always a mapping to some field, or is it possible that sometimes tensors can be given as a map from a vector space to vector space like $T:V\to W$ or $T:V^* \times V\to W$ ? ex: stress tensor(which takes in a vector and maps it to another vector.

I did read somewhere it is possible to raise/lower * and transfer $V$ or $V^*$ to the other side, but did not understand it and haven't found any material explaining the topic clearly.

Answer 1

A single tensor can be viewed in many different ways.

For example: $T:V^* \times V \to \mathbb{R}$.

Plug in a dual vector and a vector and get a scalar, so $T$ is a bilinear scalar valued map.

On the other hand let's just plug in a dual vector, say $T(v^*,\cdot)$. Now if I then plug in a vector: $T(v^*,w)$ I get a scalar (and we're linear in $w$) so $T(v^*,\cdot)$ is itself a dual vector. Thus $T$ takes in a dual vector (i.e. $v^*$) and outputs a dual vector (i.e. $T(v^*,\cdot)$). In other words, $T:V^* \to V^*$.

Similarly, we could just plug in the second slot leaving the first blank: $T(\cdot,w)$ and get an element of $V^{**}$ which is naturally identified (for finite dimensional vector spaces) with $V$. Thus $T$ takes in a vector (i.e. $w$) and kicks out a vector (actually a double dual vector) (i.e. $T(\cdot,w)$). In other words, $T:V \to V$ (after identifying $V^{**}$ and $V$).

So my example can be viewed as a bilinear map $V^* \times V \to \mathbb{R}$ or as a linear map $V^* \to V^*$ or as a linear map $V \to V$. It can also be viewed as an element of the tensor product space $(V^* \otimes V)^* \cong V \otimes V^*$.

This is understandably confusing. There are a bunch of natural isomorphisms and identifications floating around, so a single tensor can be viewed in many many ways.

The root issue we're really dealing with is currying.

Answer 2

If $V$ is a finite dimensional vetor space over a field $\Bbb F$ there is a way to relate $V$ with $V^*$ granted by the Riesz' Lemma:

If $V$ is a finite dimensional vector space with an inner product, for each covector $f\in V^*$ there exists a unique vector $a\in V$ such that $$f(v)=a\cdot v$$

The element $a$ of $V$ it is dubbed the Riesz representative of $f$.

Some mathematicians don't like the word covector, so others would prefer the name linear functional or rank one covariant tensor.