Let $X$ be a finite dimensional real Banach space and $X^*$ be the dual of $X$. I am going to calculate the dual of operator space, $L(X)$. Is it possible to conclude that the dual of $L(X)$ is $L(X^*)$?
$\require{AMScd}$ \begin{CD} X @>{T:X\to X}>> L(X)\\ @V{f:X\to \mathbb{R}}VV @VV{??(f':L(X)\to \mathbb{R})}V \\ X^* @>>{T':X^*\to X^*}> L(X^*) \end{CD}
Since, I am new in this topic I do not know much about the theories. If the statement is false then please help me with a counterexample. Also, please tell me in which case we can conclude such. Thank you in advance.