This is a question of functional analysis: Start with a locally convex space $X$, form its dual $X^*$ consisting of continuous  linear functionals on $X$. Now $X^*$  has several locally convex topologies $\tau$ with the following property:
 

>  For any $x\in X$ the linear map $L_x:X^*\to\mathbb{R}$ is $\tau$-continuous. 

For  such topology  $\tau$  we form  the bidual $X^{**}_\tau$ consisting of linear functionals on $X^*$ that are $\tau$ continuous.  From the choice of $\tau$ we deduce that we have a natral map

$$X\ni x\mapsto L_x\in X^{**}_\tau. $$

The question is if there exist topologies $\tau$ on $X^*$ such that  the above map is an isomorphism of locally convex spaces.

A natural topology on $X^*$ is the  smallest locally convex topology such that all the linear functionals $L_x$ are continuous.  Let's call this $\tau_0$. (Traditionally it is denoted by $\sigma(X^*, X)$.

We can ask a more refined   question: for which locally convex spaces the bidual $X^{**}_{\tau_0}$ coincides with $X$. 

There is know a large class of such spaces namely the nuclear spaces. For details see 

>Gelfand & Shilov: Generalized Functions., vol 2.

For example the space $X=C^\infty(M)$, $M$ compact  has this property.