I have this theorem:

The proof starts by proving that the set is linearly independent. It does this as follows:Theorem 33.2: Let and be dual bases for and . Then the set of tensor products:

forms a basis for (the set of all tensors of order on a vector space ).

"We shall prove that the set of tensor products is a linearly independent generating set for . To prove that the set is linearly independent, let

where the RHS is the 0 tensor".

Firstly, what is ? Is it a linear map going from to with the bases outlined in the question?

Secondly, how will this prove that the set is linearly independent?