Tensor products of weakly compact sets

Question

Let $X$ and $Y$ be Banach spaces. Denote by $X\otimes_\varepsilon Y$ the injective tensor product of $X$ and $Y$. Also, let $A\subset X, B\subset Y$ be any sets. Set $A\otimes B = \{a\otimes b\colon a\in A, b\in B\}$. Can we deduce that if $A$ and $B$ are weakly compact then so is $A\otimes B$ just because the bilinear map $(a,b)\mapsto a\otimes b$, $X\times Y\to X\otimes_\varepsilon Y$ is continuous?