if for every choice of them the inner product A_{μν} B^{μ} C^{ν} is a scalar, then A_{μν} is a co-variant tensor. The last law holds even when there is the more special formulation, that with any arbitrary choice of the four-vector B^{μ} alone the scalar product A_{μν} B^{μ} B^{ν} is a scalar, in which case we have the additional condition that A_{μν} satisfies the symmetry condition. According to the method given above, we prove the tensor character of (A_{μν} + A_{νμ}), from which on account of symmetry follows the tensor-character of A_{μν}. This law can easily be generalized in the case of co-variant and contravariant tensors of any rank.
Finally, from what has been proved, we can deduce the following law which can be easily generalized for any kind of tensor: If the quanties A_{μν} B^{ν} form a tensor of the first rank, when B^{ν} is any arbitrarily chosen four-vector, then A_{μν} is a tensor of the second rank. If for example, C^{μ} is any four-vector, then owing to the tensor character of A_{μν} B^{ν}, the inner product A_{μν} C^{μ} B^{ν} is a scalar, both the four-vectors C^{μ} and B^{ν} being arbitrarily chosen. Hence the proposition follows at once.
A few words about the Fundamental Tensor g_{μν}.
The co-variant fundamental tensor—In the invariant expression of the square of the linear element
ds^2 = g_{μ}{ν} dx_{μ} dx_{ν}
