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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?

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