(All of this question uses $V$ as the real vector space where tensors are being built from).
I am currently trying to learn about tensor notation, and I am running into a road block with understanding what exactly constitutes as a vector.
In the formal definition, a tensor seems to be defined bluntly as “an element of the tensor space over $V$”. For example:
- $(0,1)$-tensors are elements of $V$.
- $(1,0)$-tensors are elements of $V^*$.
- $(1,1)$-tensors are elements of $V \otimes V^*$
However now comes my confusion. When we think of a vector $v \in V$ with respect to a (say three dimensional) basis $\{e_1, e_2, e_3\}$, then in tensor notation we write $v = a^i e_i$, where $a^1, a^2, a^3$ are the coordinates in this basis.
Then $a^i$ gets called a “contravariant tensor” (in other words a $(1,0)$-tensor), but how is it a tensor with respect to the formal definition? Surely each $a^i \in \mathbb{R}$, and $a^i \notin V^*$.
The same goes for something like the metric tensor, which I see denoted as $g_{i j}$, but this is just the components and again just a bunch of elements from $\mathbb{R}$?
A contravariant tensor here means an $(1,0)$ tensor, that is, an element of $V^*$, i.e. a linear map $V\to\Bbb R$.
Specifically here, we get the elements of the dual basis of $e_1,\dots, e_n$, which are just the coordinate mappings: $\sum_ia^ie_i\mapsto a^j$.
The metric tensor is of type $(2,0)$, as it is a bilinear map $V\times V\to\Bbb R $, which are in one-to-one correspondence with elements of $V^*\otimes V^*$.