Let $X$ be a vector space. The dual is the set of all linear forms from $X$ to $\mathbb{F}$.(can be $\mathbb{R}$ or $\mathbb{C}$) That is, $X^* = \operatorname{Hom}(X,\mathbb{F})$.
If $X$ is also Banach, the continuous dual is the set of all continuous maps in $X$. That is, $C(X,\mathbb{F}) \cap X^*$.
Is there a symbol for the continuous dual space? Wikipedia uses the prime $'$, but that coincides with derived sets and other common meanings of $'$. Some books simply use $X^*$ to refer to the continuous dual space.