# Definition of tensor product of two dual spaces of two given vector spaces

Let $$X$$ and $$Y$$ be two vector spaces over $$\mathbb{C}$$ and let $$X^*$$ and $$Y^*$$ be the algebraic duals of $$X$$ and $$Y$$, respectively. By definition, $$X\otimes Y$$ is the linear span of the set $$\{eval_{(x,y)}:x\in X, y\in Y\}$$ (functionals on the space of all bilinear functionals on $$X\times Y$$).

In a text on tensor products (by Raymond Ryan), it's claimed that an element $$u=\sum_{1\leq i\leq n} \phi_i \otimes \psi_i \in X^* \otimes Y^*$$ is zero if and only if $$\sum_{1\leq i\leq n} \phi(x)\psi(y)=0,\forall x\in X,y\in Y$$. I just need a verification of this statement.

By the definition of $$X^*$$, an element $$\phi \in X^*$$ is zero if and only if $$\phi(x)=0,\forall x\in X$$. Similar definition holds for $$Y^*$$. Now my question is how these definitions of $$X^*$$ and $$Y^*$$ help us to derive the above mentioned claim. In general $$X^{**} \neq X$$. So, what am I missing?