I want to calculate the dual space of $\mathcal{C}\_0[a,b]$, that is the space of continuous functions on $[a,b]$ vanishing at $a$. I know that the dual space of $\mathcal{C}[a,b]$ (continuous functions) is the space of differences of Lebesgue-Stieltjes measure associated to increasing functions, left-continuous, and vanishing at a fixed point in $[a,b]$. Anyone can help me?



Dual space of $\mathcal{C}_0[a,b]$
Each (bounded linear) functional on $C[a,b]$ is also a functional on $C\_0[a,b]$. Each function $f \in C[a,b]$ can be written as $f(x) = f\_0(x) + f(a) \cdot 1$, where $f\_0 \in C\_0[a,b]$. On the other hand, each functional on $C\_0[a,b]$ can be extended to a functional on $C[a,b]$ (Hahn-Banach). Therefore the answer is $(C[a,b])^{\ast}/\mathbb{C}$. In terms of those functions of bounded variation, this means that we have to require those functions vanish at one more point.


