What is the most general/abstract way to think about Tensors

Question

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?

Answer 1

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.