In this short paper, we study the nuclearity for the dual operator space $V^∗$ of an operator space 𝑉. We show that $V^∗$ is nuclear if and only if $V^{∗∗∗}$ is injective, where $V^{∗∗∗}$ is the third dual of 𝑉. This is in striking contrast to the situation for general operator spaces. This result is used to prove that $V^{∗∗}$ is nuclear if and only if 𝑉 is nuclear and $V^{∗∗}$ is exact.
