Let $X$ and $Y$ be two Banach spaces and assume, if necessary, that $X^*, Y^*$ have the approximation property (but not necessarily the Radon–Nikodym property). Consider the injective tensor product $X\otimes_\varepsilon Y$. For all $f\in X^*$, $g\in Y^*$ one can form a functional $f\otimes g$ by
$$(f\otimes g)(x\otimes y)= \langle f,x \rangle \cdot \langle g, y \rangle.$$
Is the subspace $\mbox{span}\{f\otimes g\colon f\in X^*, g\in Y^*\}$ weak* dense in $(X\otimes_\varepsilon Y)^*$?