I've trouble understanding the definition of tensor products from the bases of the spaces which the operations is applied
Given two vector spaces $V$ and $ W $ over the same field, with bases $ B_V $ and $ B_W $ respectively, I understand that one can construct the tensor product of the two spaces, denoted $ V \otimes W $, and that the basis for this tensor product space is formed by the tensor products of the basis vectors of $ V $ and $ W $, like $ v_i \otimes w_j $ for $ v_i \in B_V $ and $ w_j \in B_W $.
However, I came across a description that talks about a function that maps a pair $ (v, w) $, where $ v \in B_V $ and $ w \in B_W $, to 1, and maps all other elements in the Cartesian product $ B_V \times B_W $ to 0, and this function is denoted as $ v \otimes w $. I am having trouble reconciling this description with my understanding of the tensor product, as I’ve always thought of $ v \otimes w $ as representing a specific element in the tensor product space, not a function.
Could someone clarify this function, how it relates to constructing tensor products, and how it should be properly interpreted? Additionally, any examples to illustrate this concept would be highly appreciated.
I think it is not really about tensors. We can do the same thing in the vector space $V$ alone. For every $a \in V$ there is an associated 'function' from $B_V$ to the ground field, such that its value on basis element $b_i$ is just the coefficient $\alpha_i$ in the (unique) decomposition $a = \sum \alpha_i b_i$ of $a$ with respect to this basis.