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summary:Let $X$  be a Banach space with the Grothendieck property, $Y$ a reflexive Banach space, and let $X\check{\otimes}_{\varepsilon} Y$ be the injective tensor product of  $X$ and  $Y$. \item {(a)} If either $X^{\ast \ast }$ or $Y$ has the approximation property and each continuous linear operator from  $X^\ast $ to  $Y$ is compact, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property. \item {(b)} In addition, if $Y$  has an unconditional finite dimensional decomposition, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property if and only if each continuous linear operator from  $X^\ast $ to  $Y$ is compact

The Projective and Injective Tensor Products of $L^p[0,1]$ and $X$ Being Grothendieck Spaces