considering a Tensor product $A \otimes B $ is there a standard way to construct a function that projects elements from $A \otimes B $ to $ B $ for example? Or a way to get from some generated subspace $V \subseteq A \otimes B$ the generating vectors from each one of the constituent spaces?(Lets say that $ V = \overline{ \operatorname{span} \left( a_i \otimes b_j \right)_{(i,j)}}$ reobtain the $b_j$ by some function on $V$?
Actually in my specific case, from a given set $ (v_{\alpha})_{\alpha \in I} \subseteq A \otimes B $ I believe I could simply use a basis decomposition of the $ v_{\alpha} $ in elements of $A$ and $ B $ and by the axiom of choice get the set of these elements from $B$, but for stylistic reasons a function that could take only one of the components would be more practical.