Let $T \subseteq \mathbb R$ be a closed set of real numbers. Let $X := C(T, \mathbb R)$ denote the Fréchet space of continuous real-valued functions on $T$. The topology on $X$ is generated by seminorms $\|x\|_K := \sup_{t \in K} |x(t)|$ for any compact $K \subseteq T$.

Let $X^*$ denote the dual space of continuous linear functionals on $X$. Is there a nice characterization of the dual space using the Riesz representation theorem?

Let $e_t : X \to \mathbb R$ denote the evaluation functional, defined by $e_t[x] := x(t)$ for all $t \in T$. Are the evaluation functionals dense in $X^*$?