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 $X^*$?
Best Answer
It is the space of compactly supported Radon measures. See Nicolas Bourbaki, Intégration, chapter 4, page 156 in Springer’s 2007-edition.
It seems to me that the space spanned by evaluation functionals is dense in the weak-*-topology (given any finite set of continuous functions choose—using compactness of the support—a finite open cover of the support such that in each of these open sets the given functions do not vary a lot, then use an evaluation functional for each of the open sets, at a point in the respective set), but not in the usual norm topology.