In Schutz, Geometrical Methods of Mathematical Physics, it is written
The components of a tensor are its values when it takes basis vectors and one-forms as arguments.
It then gives an abstract example of S, a (3,2) tensor, which has components
$S^{ijk}_{lm} = S(\omega^i,\omega^j,\omega^k, e_l,e_m)$
Then (same page) it says:
Prove that a general (2,0) tensor cannot be expressed as a simple outer product of two vectors. Hint: count the number of components a (2,0) tensor may have.
My question: reading what it says, I think the answer should be that there are an infinite number of components. By this, I am interpreting the definition as saying that you contract the tensor with the various vectors and one forms, resulting in a scalar, and there are an infinite number of such values that it could take on.
Obviously this is not what is meant. So, what is a "component"? Let's take the case of a (1,1) tensor in 3 dimensions, call it $M$. $M$ can be represented as 9 numbers. Also let's take a one form $w$, which can be represented as 3 numbers, and a vector v, also three numbers.
Is a component:
- the single number that is $w\, M \, v$ for some particular choice of $w,v,M$?
- the infinity of values of $w\, M\, v$ for all choices of $w,v$?
- the 9 numbers that make up $M$?
- the 6 numbers thst make up $w,v$?
- the 9+6 numbers that make up $w \, M \, v$
- ...?