# Dual of the space of continuous functions

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 $X^*$?

• Evaluation functionals are certainly not dense. At least you have to take linear combinations of them. – Gerald Edgar Oct 18 '13 at 20:20
• Related question: mathoverflow.net/q/105147/2622 – Igor Khavkine Oct 18 '13 at 20:33
• @GeraldEdgar: ha! You're absolutely right. – Tom LaGatta Oct 18 '13 at 22:02

• I don't have the book handy. Is the statement true for an arbitrary locally compact space $T$? i.e., that $C(T,\mathbb R)$ is isomorphic to the space of compactly supported Radon measures. – Tom LaGatta Oct 18 '13 at 22:13