TITLE: What is the most general/abstract way to think about Tensors
QUESTION [5 upvotes]: In their most general and abstract definitions as Mathematical Objects :

A Scalar is an element of a field used to define Vector Spaces 
A Vector is an element of a Vector Space. 

Since a Scalar is a Tensor of rank-0 and a Vector is a Tensor of rank-1, then what Space are Tensors an element of?
Can you even think of Tensors abstractly as elements of a Mathematical Space?

REPLY [4 votes]: Rank one tensors, on a vector space $V$ over the scalar field $\Bbb F$, are linear maps 
$$V\to\Bbb F$$ 
and 
$$V^*\to\Bbb F,$$ 
where $V^*$ is the dual space of $V$.
Rank two tensors are bilinear maps 
$$V\times V\to\Bbb F,$$
$$V^*\times V\to\Bbb F,$$
$$V^*\times V^*\to\Bbb F.$$
Rank three tensors are trilinear maps 
$$V\times V\times V\to\Bbb F,$$
$$V^*\times V\times V\to\Bbb F,$$
$$V^*\times V^*\times V\to\Bbb F,$$
$$V^*\times V^*\times V^*\to\Bbb F,$$
and so on.